drift effects on plasma waves
TRANSCRIPT
Drift Effects on Plasma Waves
A thesis submitted in partial fulfillment of the
requirement for the award of degree of Doctor of Philosophy in Physics
By
Muhammad Fraz Bashir Reg. No. 015-GCU-PHD-PHY-11
Department of Physics
GC University Lahore
God
Grant me serenity to accept the things,
I can’t change;
The courage to change the things, I can;
And the wisdom to know the difference!
Research Completion Certificate
It is certified that the research work contained in this thesis entitled
“Drift Effects on Plasma Waves” has been carried out by Mr.
Muhammad Fraz Bashir, Reg. No. 015-GCU-PHD-PHY-11, under my
supervision at Physics Department, GC University Lahore, during his
postgraduate studies for Doctor of Philosophy in Physics.
Supervisor
Prof. Dr. G. Murtaza National Distinguished Professor
Salam Chair in Physics
GC University, Lahore
Submitted Through
Prof. Dr. Riaz Ahmad
Chairman
Department of Physics
GC University, Lahore
Declaration
I, Mr. Muhammad Fraz Bashir, Reg. No. 015-GCU-PHD-PHY-11,
PhD scholar at the Department of Physics, GC University Lahore,
hereby declare that the matter printed in the thesis entitled “Drift Effects
on Plasma Waves” is my own work and has not been printed, or
published or submitted as research work, thesis or publication in any
form in any University/Research Institution etc. in Pakistan or abroad.
Dated: ___________ ___________________
Signature of Deponent
Acknowledgements
First of all, I say my thanks to almighty Allah for all his blessings
enabling me to complete this thesis. Secondly, I express my deep gratitude to
my supervisor Prof. Dr. G. Murtaza for his continuous support, kind guidance,
useful suggestions and moral support throughout my research period. He is
the person who laid down the foundations of plasma physics in the country
and inspired me to work in this area.
I am also extremely grateful to Prof. Dr. Andrei Smolyakov, University
of Saskatchewan (USASK), Saskatoon, Canada who first introduced me to the
Canadian Commonwealth Exchange Program Scholarship-2011 and then to
the study of Geodesic Acoustic Modes (GAMs) in the field of Tokamak Plasmas
and continuously guided and helped me till the final write-up of my thesis. Due
to his arrangements, I really enjoyed the excess to the different facilities of the
USASK like Library, Email, Super-computer lab, Gym, Swimming pool etc. My
time with him at Saskatoon specially the memorable dinner at his home, and
later through email was really fantastic!
I am also thankful to my other international collaborators for the thesis
work R. J. F. Sgalla, Prof. Dr. A. G. Elfimov, Prof. Dr. A. V. Melnikov and
Prof. Dr. M. Salimullah for their useful discussions and suggestions.
I also appreciate Prof. Dr. Peter H. Yoon who introduces me, at the end
of my PhD, to the exact numerical analysis and the quasi-linear theory using
Fortran which also results in two publications.
I am gratefully obliged to Prof. Dr. Riaz Ahmad, Chairman Physics
Department and also to Prof. Dr. H. A. Shah, ex-Chairman for providing sound
research atmosphere in the department and to Prof. Dr. Tamaz Kaladze for
several fruitful discussions.
I also thank to all my research fellows Gohar Abbas, Muhammad
Jamil, Zafar Iqbal, Hafsa Naim, Sadia Zaheer, Muddasir Ali, Tajammal
Hussain, Fazal Hadi, Naila Noreen and lab fellows Abdur Rasheed, Azhar
Hussain, Muhammad Shahid, Aroj Khan for their valuable discussions and
encouraging behavior. I would like to acknowledge Tariq Azeem, Khalid Azfal
and Muhammad Tariq from Office of Salam Chair for their help during my
research.
Special Thanks to my wife, Fatima Zafar, whose invaluable emotional
support and care helped me to maintain a resilient spirit throughout the often
difficult times during my PhD.
I am particularly grateful to my parents for their love and incessant
prayers and thankful to my affectionate brothers and sisters for their support
and encouragements. I am also grateful to my parents-in-law, brothers-in-laws
and sister-in-Law for their affection and prayers.
Finally, I gratefully acknowledge the Commonwealth Fellowship
Program of FAIT Canada and Natural Sciences and Engineering Research
Council of Canada (NSERC) for giving me six month scholarship and travel
allowance to complete part of my research work in USASK, Canada.
Muhammad Fraz Bashir
List of Publications Publications included in this thesis
1. Electromagnetic effects on Geodesic Acoustic Modes
M. F. Bashir, A. I. Smolyakov, A. G. Elfimov, A. V. Melnikov and G. Murtaza, Phys.
Plasmas 21, 082507 (2014).
2. Stability analysis of self-gravitational electrostatic drift waves for a streaming non-
uniform quantum dusty magneto-plasma.
M. F. Bashir, M. Jamil, G. Murtaza, M. Salimullah, H. A. Shah, Phys. Plasmas 19,
043701 (2012).
3. Drift effects on Geodesic Acoustic modes
R. J. F. Sgalla, A. I. Smolyakov, A. G. Elfimov, M. F. Bashir, Phys. Lett. A 377, 303
(2013).
Publications not included in this thesis
1. Relativistic Bernstein mode instability
M. F. Bashir, N. Noreen, G. Murtaza, P. H. Yoon, Plasma Phys. Control. Fusion 56,
055009 (2014).
2. On the ordinary mode instability for low beta plasmas
F. Hadi, M. F. Bashir, A. Qamar, P. H. Yoon, R. Schlickeiser, Phys. Plasmas 21,
052111 (2014).
3. Drift kinetic Alfven wave in temperature anisotropic plasma
H. Naim, M. F. Bashir and G. Murtaza, Phys. Plasmas 21, 032120 (2014).
4. Whistler Instability in a semi-relativistic Bi-Maxwellian Plasma
M. F. Bashir, S. Zaheer, Z. Iqbal, G. Murtaza, Phys. Lett. A 377, 2378 (2013).
5. Effect of temperature anisotropy on various modes and instabilities for a non-
relativistic Maxwellian Plasma.
M. F. Bashir, G. Murtaza, Braz. J. Phys. 42 , 487 (2012).
6. Perpendicularly propagating modes for a weakly magnetized relativistic degenerate
electron plasma
G. Abbas, M. F. Bashir, G. Murtaza, Phys. Plasmas 19, 072121, (2012).
7. Study of high frequency parallel propagating modes in a weakly magnetized
relativistic degenerate electron plasma
G. Abbas, M. F. Bashir, M. Ali and G. Murtaza, Phys. Plasmas 19, 032103 (2012).
8. Anomalous skin effects in relativistic parallel propagating weakly magnetized
electron plasma waves.
G. Abbas, M. F. Bashir, G. Murtaza, Phys. Plasmas 18, 102115 (2011).
9. On the drift magnetosonic waves in anisotropic low beta plasmas
H. Naim, M. F. Bashir and G. Murtaza, Phys. Plasmas 21, Accepted (2014).
Abstract
The full kinetic dispersion relation for the Geodesic acoustic modes (GAMs) including
diamagnetic effects due to inhomogeneous plasma density and temperature is derived by using
the drift kinetic theory. The fluid model including the effects of ion parallel viscosity (pressure
anisotropy) is also presented that allows to recover exactly the adiabatic index obtained in kinetic
theory. We show that diamagnetic effects lead to the positive up-shift of the GAM frequency and
appearance of the second (lower frequency) branch related to the drift frequency. The latter is a
result of modification of the degenerate (zero frequency) zonal flow branch which acquires a
finite frequency or becomes unstable in regions of high temperature gradients. By using the full
electromagnetic drift kinetic equations for electrons and ions, the general dispersion relation for
geodesic acoustic modes (GAMs) is derived incorporating the electromagnetic effects. It is
shown that m=1 harmonic of the GAM mode has a finite electromagnetic component. The
electromagnetic corrections appear for finite values of the radial wave numbers and modify the
GAM frequency. The effects of plasma pressure βe, the safety factor q and the temperature ratio τ
on GAM dispersion are analyzed. Using the quantum hydrodynamical model of plasmas, the
stability analysis of self-gravitational electrostatic drift waves for a streaming non-uniform
quantum dusty magneto-plasma is presented. For two different frequency domains i.e.,
Ω0d<<ω<Ω0i (unmagnetized dust) and ω<< Ω0d < Ω0i (magnetized dust), we simplify the general
dispersion relation for self-gravitational electrostatic drift waves which incorporates the effects
of density inhomogeneity ∇n0α, streaming velocity v0α due to magnetic field inhomogeneity ∇B₀,
Bohm potential and the Fermi degenerate pressure. For the unmagnetized case, the drift waves
may become unstable under appropriate conditions giving rise to Jeans instability. The modified
threshold condition is also determined for instability by using the intersection method for solving
the cubic equation. We note that the inhomogeneity in magnetic field (equilibrium density)
through streaming velocity (diamagnetic drift velocity) suppress the Jeans instability depending
upon the characteristic scale length of these inhomogeneities. On the other hand, the dust-lower-
hybrid wave and the quantum mechanical effects of electrons tend to reduce the growth rate as
expected. A number of special cases are also discussed.
Contents
1 Introduction 6
1.1 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Magnetically Confined Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Toroidal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Quantum Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Geodesic Acoustic Modes (GAMs) . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Jeans Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Types of Drifts in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6.1 Non-uniform B Field Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.2 Diamagnetic Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.1 Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.2 Drift Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.3 Linear Drift Kinetic Equation for Homogenous Plasma Temperature and
Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7.4 Linear Drift Kinetic Equation for Inhomogeneous Plasma Temperature
and Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.5 Two Fluid Ideal MHD Model . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.6 Quantum Magnetohydrodynamic Model . . . . . . . . . . . . . . . . . . . 22
1.7.7 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Drift Effects on Geodesic Acoustic Modes 24
1
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Drift Kinetic Equation and Full Kinetic Dispersion Relation . . . . . . . . . . . . 26
2.2.1 Density Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Quasi-neutrality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4 Simplified Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Two Fluid MHD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Electromagnetic Effects on Geodesic Acoustic Modes 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Drift Kinetic Equation and Full Kinetic Dispersion Relation . . . . . . . . . . . . 37
3.2.1 Density Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 Full Kinetic Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Simplified Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Electron Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 The Ion Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 The Dispersion Relation and Magnetic Perturbation Effect . . . . . . . . 44
4 Stability Analysis of Self-Gravitational Electrostatic Drift waves for a Stream-
ing Non-Uniform Quantum Dusty Magnetoplasma 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Dielectric Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 For Unmagnetized Dust Grains . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 For Magnetized Dust Grains . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Summary of Results and Discussion 65
6 Appendixes 69
6.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2
6.3 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.6 Appendix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3
List of Figures
1-1 Schematic of changing in the states of water by increasing the temperature . . . 7
1-2 Schematic of Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1-3 (a) The Toroidal Geometry (b) flux surfaces . . . . . . . . . . . . . . . . . . . . . 10
3-1 Multiple roots of the dispersion relation (3.45) i.e., Normalized frequencyωR
vtivsas functions of krρi at different values of plasma pressure: βe =0.05 — solid
line, βe = 0.07— dotted line, βe = 0.1 — dashed line, βe = 0.2 — dot-dashed line;
q = 1.8 and τ = 1.6.
48
3-2 Normalized FrequencyωR
vtivs krρi for GAM branch at fixed q = 1.8 and βe = 0.2
for different values of the temperature ratio: τ = 1.6 — solid line, 1.7 — dotted
line, 1.8 — dashed line, 1.9 — dot-dashed line.
49
3-3 Normalized frequencyωR
vtivs krρi for GAM branch at fixed τ = 1.3 and βe = 0.1
for different values of safety factor: q = 1.8 — solid line, 2.5 — dotted line, 3.5 —
dashed line, 4.5 — dot-dashed line.
49
3-4 Normalized frequencyωR
vtivs βe for GAM branch at fixed q = 2.5 and τ = 1.3
for different values of normalized radial wave number: krρi = 0.1 — solid line, 0.2
— dotted line, 0.3 — dashed line, 0.4 — dot-dashed line.
50
4
3-5 Normalized frequencyωR
vtivs βe for GAM branch at τ = 1.8 and krρi = 0.4 with
variation in safety factor: q = 2.0 — solid line, 2.5 — dotted line, 3.0 — dashed
line, 3.5 — dot-dashed line.
50
3-6 Normalized frequencyωR
vtivs βe for GAM branch at fixed value of q = 2.5 and
krρi = 0.3 for different values of temperature ratio: τ = 1.0 — solid line, 1.3 —
dotted line, 1.6 — dashed line, 1.9 — dot-dashed line.
51
3-7 Normalized frequencyωR
vtivs τ for GAM branch at fixed q = 3.0 as a function of
τ : a) βe = 0.1, Eq.(3.47) — solid line, krρi = 0.05 — double dotted, krρi = 0.1 —
dotted line, krρi = 0.2 — dashed line, krρi = 0.4 — dot-dashed line; b) krρi = 0.1,
Eq. (3.47) — solid line, βe = 0.1 — double dotted, βe = 0.2 — dotted line, βe = 0.3
— dashed line, βe = 0.4 — dot-dashed line.
51
3-8 Normalized frequencyωR
vtivs q for GAM branch at fixed τ = 1.6 as a function of
q, a) βe = 0.03, Eq. (3.47) — solid line, krρi = 0.2 — double dotted, krρi = 0.3 —
dotted line, krρi = 0.4 — dashed line, krρi = 0.5 — dot-dashed line; b) krρi = 0.1,
Eq. (3.47) — solid line, βe = 0.05 — double dotted, βe = 0.1 — dotted line, βe = 0.2
— dashed line, βe = 0.4 — dot-dashed line.
52
4-1 This figure represents the graph of f(ω) vs ω for Eq. (4.15). Solid curve is
for f(ω) = ω3 and dotted curves are for f(ω) = B ω2 − ω C + G for set of
parameters given after Eq. (4.16) with variation of streaming velocity as (i)
dotted for v0 = 10−7c, (ii) small dashed for v0 = 3× 10−7c (iii) large dashed for
v0 = 5× 10−7c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5
Chapter 1
Introduction
1.1 Plasma
The word "plasma" is a Greek word meaning “formed” or “molded” introduced by Czech phys-
iologist named "Jan Evangelista Purkinje" in the mid-19th century [1]. When any substance is
heated to such temperatures overcoming the binding energy of that particular state, the phase
transition occurs. Fig.(1.1) depicts the transition between different states of water by increasing
the temperature, the corresponding states of matter and some of the types of plasmas occur in
these ranges [2]. The path to the fourth state of matter, which is called plasma, starts from
first state ice (solid) then ice on heating becomes water (liquid-second state), water becomes
steam (third state) and finally to the ionized gas (fourth state). Thus the plasma is the ionized
quasi-neutral gas in which the thermal energies are greater than the interaction energies and
the interaction between the charged particles is predominantly collective as many particles,
through their long range electromagnetic fields, interact simultaneously.
In literature, the different types of plasma are available depending upon the different species
in it. The most common case is the electron-ion plasma and other cases are the electron-positron
plasma, electron-hole plasma (semiconductor plasma), electron-positron-ion plasma and the
electron-ion-dust plasma. The existence of neutral atoms determines the degree of ionization
of plasma i.e., if neutral atoms are present, it is called partially ionized plasma otherwise fully
ionized plasma. In this thesis, we will consider the fully ionized case for electron-ion plasma
and the electron-ion-dust plasma called as dusty.
6
Figure 1-1: Schematic of changing in the states of water by increasing the temperature
The basic plasma parameters are the temperature T , the density n0 and the magnetic field
B0 usually measured in electron-Volt (eV ), particles per cubic centimeter (cm−3) and Gauss
(G) respectively. These three parameters give birth to the three subsidiary parameters i.e.,
the thermal velocity vtα =
γkBTα/mα, the plasma frequency ωpα =
4πn0e2α/mα and the
cyclotron frequency ωcα = eαB0/mαc where γ is the adiabatic index, kB is the Boltzmann
constant, eα is the charge, mα is the mass and c is the velocity of light and α is designated for
different species in plasma. The ratio of different combinations of these subsidiary parameters
leads to the well-known derived quantities the Debye length λDα=
Tα/4πn0e2α, Larmor
radius rLα = vtα/ωcα and the plasma beta βα = 8πn0Tα/B20 . The derived quantities are used
to characterize the different plasma environments.
The plasma variables are the wave frequency ω and the wave vector k which determine
the time scale and the length scale of a phenomenon under observation. The wave frequency
ω (wave vector k) articulates the time scale (length scale) is fast or slow (long or short) as
compared to the characteristic frequency (length) of the particles i.e., plasma frequency or the
cyclotron frequency (Debye length or Larmor radius). The ratio of the wave frequency to the
wave vector (ω /k = vφ ) defines phase speed of the wave and can be compared with the thermal
7
speed vtα of particular species to analyze the dispersion characteristics of the wave.
1.2 Magnetically Confined Plasma
The demand of the worldwide energy is increasing day by day. Saving energy and using re-
newable energy sources are not enough to fulfill this increasing demand. Nuclear energy on the
other hand is obtained by two types of processes i.e., fission and fusion. The fission is a process
in which the energy is produced when the heavy elements like uranium break-up and the fusion
gives energy when the lights elements like deuterium and tritium merge together . The topic
of interest for this thesis is fusion since controlled fusion becomes more realistic.
The fusion is also divided into two categories called inertial confinement fusion (ICF) and
the magnetically confined fusion (MCF). In former approach, the confinement of the plasma
is achieved by making use of petta-watt laser with energy greater than 10MJ to irradiate,
uniformly, a solidified micro target of spherical shape while the latter approach deals with
the confinement due to the very strong magnetic field generated by the superconductor coils.
Both approaches tend to satisfy the Lawson criterion differently i.e., in MCF approach, the
confinement is achieved for long time but low densities whereas ICF confines the plasma for
short time at extremely high densities. The table below depicts the comparison of parameters
for these two types of confinement.
MCF ICF
Particle Density ne 1014cm−3 1026cm−3
Confinement time τ 10 s 10−11s
Lawson criterion neτ 1015scm−3 1016scm−3
As described above that the in MCF approach, the suitable magnetic field configuration
can confine the all charged particles in plasma. A magnetic device, which is called tokamak,
containing a vacuum is used to do so in which mixed tritium and deuterium are inserted. The
schematic picture of tokamak with the magnetic configuration is shown Fig (1.2) portraying
the poloidal and toroidal magnetic fields configuration by passing the current in the coils and
the net magnetic field. The contact of plasma with the walls is avoided due its high temper-
ature requirement by restricting the movement of the particles perpendicular to the field lines
8
Figure 1-2: Schematic of Tokamak
but allowing free movements in the longitudinal direction. Thus the suitable magnetic field
configuration leads to motion of the particles in closed orbits and never escapes.
The design and operation of the Spherical tokamaks are very similar to the tokamaks but
the aspect ratios of both machines are the actual difference where the aspect ratio is the ratio
of the major and minor radii. This ratio for the tokamaks is about between 3 to 5 whereas for
spherical tokamak it is less than 1.5. This less aspect ratio allows spherical tokamaks to operate
at high value of the magnetic pressures that leads to achieve the better confinement. Now a
days, there are running two or more experiments on the spherical tokamaks in Culham, UK ,
New Jersey, USA and Saint Petersburg, Russia the name of the devices are the Mega-Ampere
Spherical Tokamak (MAST), the National Spherical Tokamak Experiment (NSTX) and the
Globus-M tokamak respectively.
1.2.1 Toroidal Geometry
The axisymmetric tokamak geometry can be visualized as simple (quasi-) toroidal coordinates
r, θ, ς, where r is the minor radius, θ and ς are poloidal and toroidal angles [3]. As shown in
9
Figure 1-3: (a) The Toroidal Geometry (b) flux surfaces
Fig.(1.3 a,b) to specify the position on the coordinate surface, one should find the intersection
of two curves from each grid at coordinate surface, r = constant. The property of the curves
of the grid is that any two curves of same family never intersect each other but each curve
from one family crosses the every curve of other family precisely at one point. Thus on the
toroidal surfaces, the every point is labeled as θ + 2πm and ς + 2πn where m and n are
called the toroidal and poloidal mode numbers. The safety factor parameter q(r) is defined as
q(r) ≡ r/R(Bς/Bθ(r)) i.e., the ratio of poloidal magnetic field Bθ(r) and the toroidal magnetic
field Bς(r = 0) where r and R are the minor and major radii respectively. The safety factor
gives the qualitative analysis about the stability of plasma in tokamaks that how many numbers
of toroidal rotations are necessary for a single poloidal rotation of magnetic field line. The value
of the safety factor at the edge of the plasma is qmin(r) > 2− 3. The magnetic field, in the low
pressure plasma where the magnetic surfaces are supposed to be concentric, is
B =Bςeς +Bθeθ
1 + ǫcosθ(1.1)
where eς , eθ are the unit vectors in poloidal and toroidal directions and ǫ = r/R is the inverse
aspect ratio.
10
1.3 Quantum Plasma
In contrast to the classical plasma which is characterized by the high temperature and low
density, Quantum plasmas are described by the low temperature and the high density. The
thermal de Broglie wavelength of the particles i.e.,
λB =
mvT, (1.2)
depending on its small or large value as compared to the inter-particle distance defines
whether the plasma is classical or Quantum. In classical plasmas, the particles are considered
as point like due to their small de Broglie wavelength as a result of that no quantum interference
occurs and the wave functions do not overlap. The quantum effects on the other hand become
prominent when the average interparticle distance n−1/3 becomes equal or smaller than the de
Broglie wavelength i.e.,
nλ3B ≥ 1. (1.3)
The statistical mechanics reveals that the quantum effects become important in the case of
ordinary gases when the Fermi temperature becomes greater than the thermal temperature.
The Fermi energy and the Fermi temperature are defined as
EF ≡ kBTF =2
2m
3π2n
2/3. (1.4)
From the above expression of Fermi temperature, a dimensionless parameter χ which the
ratio of thermal temperature to the Fermi temperature can be related to the dimensionless
parameter nλ3B as
χ =TFT
=1
2
3π2
2/3 nλ3B
2/3. (1.5)
When the dimensionless parameter χ ≥ 1, quantum effects become important. In quantum
plasma, the thermal velocity analog is the Fermi velocity and the Debye length analog is the
Fermi screening length which are given by
11
vF =
2kBTF/m =
m
3π2n
1/3(1.6)
λF =vFωp
. (1.7)
Quantum plasmas have attracted much attention due to their applications in electronic
devices having ultra-small size [4], laser plasmas [5] and dense astrophysical plasmas [6]-[8].
Recent developments in quantum plasmas include quantum beam instabilities modification
with quantum corrections [9]-[12], quantum drift waves [13], quantum ion-acoustic waves [14],
Debye screening modifications with magnetic fields for quantum plasmas [15] etc.
1.4 Geodesic Acoustic Modes (GAMs)
Geodesic Acoustic Modes (GAMs) are linear eigen-modes of poloidal plasma rotation supported
by plasma compressibility in toroidal geometry and were first predicted by Winsor et al., [16].
GAMs couple to the drift waves linearly via the toroidal side bands of the plasma perturbations
([16]-[20]) and contain the anisotropic pressure perturbations. The different types of geodesic
oscillations with the wide range of frequencies are observed in different tokamaks ([21]-[24]) and
also in stellarators [25].
The different theoretical models predict different forms of dispersion relation for GAMs
asuuming that any perturbation quantity χ is assumed the form χ = χm exp(−imθ + inζ −iωt)+χm±1 exp(i(m±1)θ+inζ−iωt). The ideal Magnetohydrodynamic (MHD) used by Winsor
et al., [16] results in ωGAM = (
2γ(Te + Ti)/mi)/R where γ represents the ratio of specific
heat for same response of electron and ions and R is the major radius. The two fluid model due
to different response of electrons and ions gives different coefficients of Te and Ti [26]. When
the kinetic theory is used, the dispersion relation becomes ωGAM = (
2(Te + 7Ti/4)/mi)/R.
Smolyakov et al., [27] by including the effect of anisotropy of plasma pressure through the
parallel viscosity tensor, retrieved the same result of kinetic theory from MHD model. The
standard GAM involves m = n = 0 for electric field φ0 and the m = 1 component of plasma
density perturbations i.e., n±1. However when we take finite m and n, the GAM gets modified
12
by Alfven wave as
ω2GAM =7
4
v2tiR2
+ k20v2A ; k0 =
m− nq
R. (1.8)
In the same letter, they also show the electromagnetic dispersion relation of GAMs with different
polarizations and their coupling to Alfven branch as
ω2GAM =21
8
v2tiR2
+v2A
q2R2, (1.9)
and
ω2GAM =7
8
v2tiR2
+v2A
q2R2. (1.10)
1.5 Jeans Instability
The galaxy consists of giant molecular clouds which remain stable due to balance between the
self-gravity and the pressure due to finite temperature. When there is a misbalance between
the inward pressure of gas and the gravity, the collapse of interstellar clouds occurs and the
star formation happens and the perturbation grows exponentially. This is so called the Jeans
instability on the name of Sir James Jeans. The perturbation theory for electrostatic waves
gives the dispersion relation of sound waves as
ω2 = k2v2s , (1.11)
showing that the sound wave is non-dispersive. As the effect of gravity is included the dispersion
relation modifies to
ω2 = k2v2s − ω2J ; ω2J = 4πGρ0, (1.12)
and the dispersion effects enter in the sound wave. However if we compare the above dispersion
relation to the dispersion relation of electron plasma waves
ω2 = k2v2te + 4πn0e2/me, (1.13)
the main difference between Eqs.(1.12, 1.13) resides in the sign that the plasma waves are always
stable but the Eq.(1.12) shows that it becomes purely growing if the last term becomes bigger
13
than the first term and we obtain the growth rate of Jeans instability as
γ =
ω2J − k2v2s , (1.14)
in which the gravitational effects are greater than the inward pressure.
The presence of charged dust grains is the most important phenomenon in astrophysical
situations. It is believed that the dust being the prominent constituent of cosmic environment
and may explain the different type of processes in the astrophysical compact objects. The self-
gravitational instability in the dusty plasma has been discussed by many authors in the recent
years and the new criteria about this instability was obtained. In particular, Salimullah et al.,
[28] discussed the Jeans instability for dust lower hybrid case and also include the quantum
effect through Bohm potential as
γ =
ω2Jd −ω2dlhfi
−ω2Qd4
, (1.15)
where
ω2Jd = 4πGρd , ω2dlh =ωpdωciωpi
, fi = 1−ω2Qd4ω2dlh
; ω2Qd =2k4
m2d
In this thesis, we will extend the work of Ref. [28] with modification due to the drift effects
i.e., the density gradient and the magnetic field gradient.
1.6 Types of Drifts in Plasmas
The intuitive and quite accurate analytic solutions of the velocity (drift) of charged particles can
be determined in arbitrarily complicated electric and magnetic fields which are slowly changing
in both space and time. Drift solutions are obtained by solving the Lorentz equation
mα∂vα∂t
= qα (E+ vα ×B) . (1.16)
The magnitude of this steady drift is easily calculated by assuming the existence of a constant
perpendicular drift velocity in the Lorentz equation, and then averaging out the cyclotron
14
motion:
E+ vα ×B = 0. (1.17)
Taking cross product of above equation with B, we get
VE = vα =E×BB2
. (1.18)
This is called "E-cross-B" drift. The general force drift velocity can easily be derived from
E ×B drift by simply making the replacement E → F/q in the Lorentz equation as
VFα =Fα ×BqαB2
. (1.19)
Comparison of the Equations 1.18 and 1.19 leads to two important conclusions. One is that
a bulk motion, with the velocity VE, of the entire plasma across the magnetic field occurs due
to the electric field which is perpendicular to the magnetic field but does not drive currents in a
plasma. On the other hand, a cross field current is driven by the force (i.e., gravity, centrifugal
force, etc.) perpendicular to magnetic field due to the motion of charged particles (electrons
and ions) in opposite directions.
1.6.1 Non-uniform B Field Drift
For uniform magnetic field, the exact expressions for the drift velocity are derived above however
for the inhomogeneous magnetic field, with respect to space or/and time, the orbit theory is
used. In orbit theory, the ratio of Lamor radius rLα(= v⊥mα/ |qα|B) to the scale length of
homogeneity i.e., rL/L is assumed to be small and expanded.
Grad-B Drift
For case when the field lines are straight but the density changes. The gradient in the magnetic
field then leads to the variation in the Larmor radius i.e., Larmor radius will decrease as
magnetic field strength will increase and vice versa. The grad-B drift is perpendicular to both
the B and ∇B and is proportional to the rL/L and v⊥[29] is given by
15
V∇B = ±1
2v⊥rLα
B×∇B
B2. (1.20)
The coefficient 1/2 comes from average and the ± is the sign of charge showing that the grad-B
drift is in opposite directions for electrons and ions and causes a current perpendicular to the
magnetic field.
Curvature Drift
Considering the case when the magnetic field lines are curve with the constant radius of cur-
vature Rc and the magnitude of magnetic field does not vary. The particles feel, while moving
with random velocity v2 along the magnetic field, the centrifugal force as given by
Fcf =mv2Rc
r =mv2R2c
Rc, (1.21)
and corresponding drift velocity becomes
VR =mαv2qαB2
Rc ×BR2c
. (1.22)
When the magnetic field lines are curve and the magnitude is also varying, like in the case
of tokamak plasma, both the grad B drift and the curvature drifts is added and given the total
drift of the particles as
VR =mα
qα
Rc ×BB2R2c
v2 +
v2⊥2
. (1.23)
1.6.2 Diamagnetic Drift
If the plasma is non-uniform due to gradient either in density or temperature or in both, the
drift velocity may be rewritten by replacing the general force with the pressure force −∇P =
−∇(nγkBT ) as
Vdia,α =B×∇PαnαqαB2
. (1.24)
This drift does not appear in single particle theory and describes the collective behavior
of plasma due to the reason that this drift involves the pressure, an average variable of the
16
plasma, obtained by taking the moment of the distribution. Both the density gradient and the
temperature gradient give rise to non-zero average transverse velocity. Due to diamagnetic drift
an effective drift current flows in the plasma as the motion of the different charged fluids are in
opposite directions to each other.
1.7 Theoretical Model
In this thesis, the different theoretical models are used to study the drift effects on plasma
waves which are given below.
1.7.1 Vlasov Equation
Each particle at any given time has a specific position and velocity. We can therefore charac-
terize the instantaneous configuration of a large number of particles by specifying the density
of particles at each point (x, v) in phase-space. The distribution function for α-species denoted
by fα(x,v, t) prescribes the instantaneous density of particles in phase-space. The number of
particles becomes fα(x,v, t)dxdv at time t having positions and velocities in the range between
x and x+ dx and v and v+ dv respectively. Thus the rate of change of the number of particles
gives the three dimensional Vlasov equation
dfαdt
=∂fα∂t
+ v · ∂fα∂x
+ a · ∂fα∂v
= 0, (1.25)
where the acceleration a from Lorentz force takes the form a =(eα/mα) (E+ v×B/c).
Thus, the distribution function fα(x,v, t) as measured when moving along a particle trajectory
(orbit) is constant. The Vlasov equation is used for collisionless plasma and together with the
Maxwell’s equations
∇.E = 4π
α
ρα , (1.26)
∇.B = 0 , (1.27)
c∇×E = −∂B
∂t, (1.28)
c∇×B =∂E
∂t+ 4π
α
Jα, (1.29)
17
to study the dynamics of electrostatic and electromagnetic plasma systems. When we use
the kinetic theory then the charge and the current densities are expressed as moments of particle
distribution fα as
ρα(x, t) = eα
fα(x,v, t)dv, (1.30)
Jα(x, t) = eα
vfα(x,v, t)dv. (1.31)
The Vlasov equation is a non-linear equation however for this thesis, the focus is only
to linear study of waves in which the field amplitude is sufficiently small. From the Vlasov
equation, the drift kinetic equation and the relativistic Vlasov equation will be formulated.
The drift kinetic equation will further be divided into two regimes i.e., (i) uniform temperature
and density case and (2) non-uniform density and temperature case.
1.7.2 Drift Kinetic Theory
In an electromagnetic field which is varying slowly in time and space, the behavior of a plasma
component can be described by the drift kinetic equation. In Vlasov equation, the distribution
function depends on the velocity vector v whereas the drift kinetic equation is dependent only
on v⊥ and v (and also, certainly, on the coordinates). Thus the drift kinetic equation for the
case of a non-stationary magnetic field and a finite electric field is of the form [30]
∂fα∂t
+∂x
∂t· ∂fα∂x
+∂v∂t
∂fα∂v
+∂v⊥∂t
∂fα∂v⊥
= 0, (1.32)
where
∂x
∂t= e0v +Vd +VE, (1.33)
∂v∂t
=eαmα
E +v2⊥2∇.e0 + vVE .∇e0, (1.34)
∂v⊥∂t
= −vv⊥
2∇.e0 +
v⊥2B
∂B
∂t+VE .∇B
, (1.35)
18
and eα, mα are the charge and the mass of α-species, e0 = B/B with B is the total magnetic
field , E = e0.E is the parallel electric field, Vd = (e0/ωcα)×v2 (e0.∇)e0 +
v2⊥/2
∇ lnB0
is the particle drift velocity due to curvature and inhomogeneity in the magnetic field, VE =
cE×B/B2 is drift velocity called "E×B drift".
Transforming the above drift kinetic equation in terms of energy per unit mass (ε =v2⊥ + v2
/2) and the magnetic moment (µ = v2⊥/2B) and correspondingly rewriting the dis-
tribution function as function of x, t, ε and µ i.e., fα(x, t, ε, µ), we obtain [30]
∂fα∂t
+∂x
∂t·∇fα +
∂ε
∂t
∂fα∂ε
= 0, (1.36)
with
∂x
∂t= e0v +Vd +VE, (1.37)
∂ε
∂t=
eαmα
vE +v2⊥2B
∂B
∂t+VE .∇B
+ v2VE.∇e0, (1.38)
∂µ
∂t= 0. (1.39)
We may linearize the above drift equation (1.36-1.39) to obtain the perturbed distribution
function fα in terms of equilibrium distribution Fα for the large aspect ratio (ǫ = r/R << 1
where r,R are minor and major radii ) systems by assuming magnetic field B = B0(1− ǫ cos θ),
which in turns gives lnB0 = (cos θ er + sin θ eθ), and neglecting the small term B = B.B0/B0
as
dfαdt
+
VE + v
B⊥
B0
.∇Fα +
eαmα
E +
v2⊥2
+ v2
VE .∇ lnB0
∂Fα∂ε
= 0, (1.40)
with making use of the following relations
19
d
dt=
∂
∂t+ v∇ +Vd.∇, (1.41)
Vd =
v2⊥2
+ v2
e0 ×∇ lnB0
ωcα, (1.42)
∇.VE = −2VE .∇ lnB0, (1.43)
e0.∇VE = −VE.∇e0 = −2VE .∇ lnB0. (1.44)
It is also convenient to write down the perturbed electric field, magnetic field and the current
density in terms of scalar potential φ and/or vector potential as
E = −∇φ− 1
c
∂A
∂t, B =∇×
Ab
and J =c
4π∇2⊥A, (1.45)
where b = e0 is the unit vector along the magnetic field. Due to considering the perturbations
with finite m and n numbers, the parallel wave-vector for the principal component and the side
bands are defined as k0 = (m− nq)/qR and k± = (m± 1− n)/qR respectively. The principal
component is assumed to be resonant inside the plasma column and considered mode dynamics
sufficiently close to the rational surface so that k± > k0.
1.7.3 Linear Drift Kinetic Equation for Homogenous Plasma Temperature
and Density
For homogeneous plasma temperature and the density, the second term in Eq.(1.40) vanishes
and the drift equation with the use of the relations in Eqs.(1.41-1.45) take the form[31]
fα = −eαφ
TαFα + gα, (1.46)
where the g distribution function satisfies the equation
ω − ωdα − kv
gα = ωJ20 (zα) (φ−
vcA)
eαFαTα
. (1.47)
20
Here
ωdα = −v2⊥/2 + v2
ωcα
krR
sin θ = − ωdα sin θ, ωdα = ωdαv2⊥/2 + v2
v2tα,
ωdα =krv
2tα
Rωcα, ωcα =
eαB0mαc
, v2tα =2Tαmα
and zα =krv⊥ωcα
.
1.7.4 Linear Drift Kinetic Equation for Inhomogeneous Plasma Temperature
and Density
Adopting the similar procedure as for the homogeneous case, the drift kinetic equation for the
inhomogeneous plasma temperature and density becomes
fα = −eαφ
TαFMα + gα, (1.48)
and the g distribution function (1.47) is modified as
ω − ωdα − kv
gα = J20 (zα)
ω − ω∗α
∂
∂θ
(φ−
vcA)
eαFMα
Tα. (1.49)
where all other quantities are similar as define in Eq.(1.47) except
ω∗α = ω∗α
1 + ηα
v2
v2tα− 3
2
; ω∗α =
cTαN ′0α
eαN0αB0r=
v2tα2Lnαωcαr
ηα = Ln/LT with scale lengths Ln = d lnN/dr and LT = d lnTα/dr.
1.7.5 Two Fluid Ideal MHD Model
The set of equations for ions and electrons including the effect of viscosity is given by
∂n
∂t+∇. (nVE) = 0, (1.50)
3
2
dp
dt+ v.∇p
+
5
2p∇.v+∇.q = 0, (1.51)
dπdt
+ p
−2v.∇ lnB − 2
3∇.v
+
2
5
−2q.∇ lnB − 2
3∇.q
= 0, (1.52)
mndv
dt+∇p+∇.π − eαn(E+ v×B) = 0. (1.53)
21
In above ideal MHD model, we note that the Eq. (1.50) is related to mass conservation,
Eq. (1.51) is for energy conservation and Eq. (1.53) represents momentum conservation. q is
used for the heat flux and π is the parallel component of viscosity tensor π as derived in Ref.
[32].
This model has been used for homogeneous plasma temperature and density by Smolyakov et
al., [27] to explained discrepancy between the results of kinetic and fluid calculations. However
in this thesis, this model is used to explain the discrepancy between the results of kinetic and
fluid calculations for inhomogeneous plasma temperature and the density.
1.7.6 Quantum Magnetohydrodynamic Model
The quantum magnetohydrodynamic model including the effect of gravitational force is given
by [28]
mαnα
∂vα∂t
+ (vα.∇)vα
= nαqα
E+
1
cvα ×B
−∇PT −∇PF
+nα
2
2mα∇
∇2√nα√nα
−mαnα∇ψ, (1.54)
and∂nα∂t
+∇ . (nαvα) = 0. (1.55)
Poisson’s equations for the electrostatic potential φ and gravitational potential ψ are
∇2φ = −4π
α
qαnα and ∇2ψ = 4πGmαnα respectively. (1.56)
Salimullah et al., [28] used this model to study the homogeneous plasma case. However in
this thesis, the above model will be used for inhomogeneous plasma to study the electrostatic
self-gravitational drift waves for magnetized electron-ion-dust plasma (i.e., α = e, i, d ).
1.7.7 Layout
The second chapter includes the effects of density and the temperature inhomogeneities on the
GAMs by using both the drift kinetic equation and the extended magnetohydrodynamic model.
22
By using the full electromagnetic drift equations for electron and ions, the dispersion effects
of Geodesic acoustic mode for m=1 harmonic is discussed in Chapter 3. Chapter 4 contains
the stability analysis of self-gravitational electrostatic drift waves for a streaming nonuniform
quantum dusty magnetoplasma. Summary of Results and discussion is presented in Chapter 5.
Appendixes are given in Chapter 6.
23
Chapter 2
Drift Effects on Geodesic Acoustic
Modes
The full kinetic dispersion relation for the Geodesic acoustic modes(GAMs) including diamag-
netic effects due to inhomogeneous plasma density and temperature is derived by using the drift
kinetic theory. The fluid model including the effects of ion parallel viscosity (pressure anisotropy)
is also presented that allows to recover exactly the adiabatic index obtained in kinetic theory.
We show that diamagnetic effects lead to the positive up-shift of the GAM frequency and ap-
pearance of the second (lower frequency) branch related to the drift frequency. The latter is
a result of modification of the degenerate (zero frequency) zonal flow branch which acquires a
finite frequency or becomes unstable in regions of high temperature gradients.
2.1 Introduction
Geodesic Acoustic Modes (GAMs) [16] is a high frequency branch of Zonal Flow (ZF) and its
role in explaining the suppression of anomalous transport and the turbulence in tokamaks has
been discussed by Diamond et al., [33]. Recently, the several authors studied different aspects
of the GAMs [34]-[36]. The dispersion relation for the high safety factor i.e., q >> 1 takes the
form ωgam = (vti/R)
Γ1 + Γ2/τ +O(q2) where τ = Ti/Te is the temperature ratio of ions to
electrons, the major radius and the ion thermal velocity is represented by R and vti =
2Ti/mi
respectively. The two adiabatic constants (Γ1,Γ2) give different coefficients depending upon the
24
models used. The ideal MHD model for one fluid [16] yielded Γ1 = Γ2 = 5/3. Kinetic models
[34],[35],[18] have predicted Γ1 = 7/4 and Γ2 = 1. Smolyakov et al., [31] showed that in GAMs,
the pressure perturbations exhibit the intrinsically anisotropic behavior and on including this
pressure anisotropy through the parallel component of ion viscosity, both the coefficients for
ideal MHD case becomes exactly similar to the coefficient resulted by the kinetic model.
Standard GAM includes symmetric poloidal perturbation (M = 0) of electrostatic potential
which couples with the poloidal density perturbation of the side-bands (M = ±1). Poloidally
inhomogeneous perturbations effectively couple to gradient driven drift wave turbulence. Beta
Alfvén Eigenmodes (BAEs )[37] are another type of GAMs with finite M = 0 due to the effect
of the gradients in the density and temperature profile. BAE dispersion relations were discussed
in local approximation [17]—[39]and also by means of ballooning theory [18],[40],[41]. GAM and
BAE have similar dispersion relations [35],[42] with the difference residing in the poloidal and
toroidal modes (M = N = 0 for GAM and M,N = 0 for BAE). BAE and related modes have
been studied in a number of papers. In particular, unstable beta induced temperature gradient
(BTG) modes have been found using MHD [38]and kinetic [39] models. Beta-induced Alfvén and
Alfvén-Acoustic Eigenmodes (BAE and BAAE, respectively) have been widely investigated due
to its role in background turbulence, generation of zonal flows and the possibility of application
in MHD spectroscopy to diagnose safety factor profiles, q(r), in tokamaks [36],[22]-[44].
Much work in studies of GAM/BAE was done with kinetic theory, particularly, taking
into account the resonance damping due to Landau mechanism [45],[46]. Yet, the fluid theory
remains an attractive alternative to full kinetic studies, especially, for studies of nonlinear
effects, e.g. nonlinear generation of GAM due to drift wave turbulence [35],[47],[48]. Much of
such work was done in fluid theory and it is desirable to have a fluid theory that self consistently
includes drift effects in both GAM modes as well in the background drift wave turbulence.
This chapter presents both the drift kinetic theory and two fluid MHD theory investigating
the diamagnetic effects on GAM. We provide a simple analytical model for GAM in presence of
diamagnetic effects. We show that in presence of diamagnetic effects, previously degenerated
(zero frequency) mode, normally associated with zonal flow acquires a finite frequency. More-
over, for large gradients of the temperature, the latter mode becomes unstable in the regions
of high q.
25
2.2 Drift Kinetic Equation and Full Kinetic Dispersion Relation
The perturbed distribution function (1.49) for the electrostatic case reduces to
fα = −eαFMTα
φ+ g, (2.1)
and g distribution takes the form
ω − k0v
g0 − ωdαg =
ω − ω∗α
∂
∂θ
φ0J
20 (zα)
eαFMTα
, (2.2)
ω − k0v
g − ωdαg0 − (ωdαg − ωdαg)− kvg =
ω − ω∗α
∂
∂θ
φJ20 (zα)
eαFMTα
, (2.3)
with
ωdα = −v2⊥/2 + v2
ωcα
krR
sin θ = −ωdα sin θ, ω∗α = ω∗α
1 + ηα
v2
v2tα− 3
2
,
ω∗α =cTαN
′0
eαN0B0R=
vtα2Lnωcαr
, ωcα =eαB0mαc
,and zα =k⊥v⊥ωcα
.
Using perturbations of the form for GAMs as
X = X0 +X1eiθ +X−1e
−iθ = X0 +X1c cos θ + iX1s sin θ, (2.4)
Eq. (2.2) becomesω − k0v
g0 +
iωdα2
g1s = ωφ0J20 (zα)
eαFMTα
, (2.5)
and after separating the different parities (g1 + g−1) and (g1 − g−1), Eq. (2.3) takes the form
ω − k0v
(g1c)−
vqR
(g1s) = [ωφ1c − ω∗αφ1s] J20 (zα)
eαFMTα
, (2.6)
and
−vqR
g1c +ω − k0v
g1s − iωdαg0 = [ωφ1s − ω∗αφ1c]J
20 (zα)
eαFMTα
, (2.7)
where we have used
ωdαg = −iωdα2
g1s,
26
and introduced the notations X1c,s = X1 ±X−1.
The solutions of Eqs. (2.4-2.6) enable to find the distribution function g
g0 = −J20 (zα)
eαFMTα
W
iωdαω
2
φ1s −
ω∗αω
φ1c
+
iωdαv2qR
φ1c −
ω∗αω
φ1s
−ω2 −
v2q2R2
φ0
, (2.8)
g1c = −J20 (zα)
eαFMTα
W
−ω2 − ω2
dα
2
φ1c −
ω∗αω
φ1s
−
vω
qR
φ1s −
ω∗αω
φ1c
−iωdαvqR
φ0
, (2.9)
g1s = −J20 (zα)
eαFMTα
W
−ω2
φ1s −
ω∗αω
φ1c
−
vω
qR
φ1c −
ω∗αω
φ1s
− iωωdαφ0
, (2.10)
and resultantly, the perturbed distribution functions of principal and side-band component
become
f0 = −eαFMα
Tα
φ0 +
J20 (zα)
W
iωdαω
2
φ1s −
ω∗αω
φ1c
+
iωdαv2qR
φ1c −
ω∗αω
φ1s
−ω2 −
v2
q2R2
φ0
,
(2.11)
f1c = −eαFMα
Tα
φ1c +
J20 (zα)
W
−ω2 − ω2
dα
2
φ1c −
ω∗αω
φ1s
−
vω
qR
φ1s −
ω∗αω
φ1c
−iωdαvqR
φ0
,
(2.12)
f1s = −eαFMα
Tα
φ1s +
J20 (zα)
W
−ω2
φ1s −
ω∗αω
φ1c
−
vω
qR
φ1c −
ω∗αω
φ1s
− iωωdαφ0
,
(2.13)
where
W = ω2 −v2
q2R2− ω2dα
2.
2.2.1 Density Perturbations
Now we calculate the density perturbation from the moment of the perturbed distribution as
nα =
fαd
3v. (2.14)
27
By using the Eqs. (2.11-2.13) in above equation, we get
n0α = −eαN0αTα
1− Γ0α −
ω2dα2ω2
I20α
φ0 +
i
2
ωdαω
I10α φ1s −
ω∗αω
I10∗αφ1c
, (2.15)
n1c = −eαN0αTα
1− I00α − ω2dα
2ω2I20α
φ1c +
ω∗αω
I00∗α −
ω2dα2ω2
I20∗α
φ1s
, (2.16)
n1s = −eαN0αTα
1− I00α
φ1s +
ω∗αω
I00∗αφ1c − iωdαω
I10α φ0
. (2.17)
The all the integrals are given in the Appendix. A.
2.2.2 Quasi-neutrality condition
The quasi-neutrality condition ( i.e., ne−ni = 0) for each component of the density perturbation
leads to the following set of equations
S01φ0 −i
2
ωdiω
S03φ1s +i
2
ωdiω
ω∗iω
S∗03φ1c = 0, (2.18)
φ1c −ω∗iω
S∗00S00
φ1s = 0, (2.19)
S′00φ1s +ω∗iω
S′∗00φ1c − iωdiω
S03φ0 = 0, (2.20)
with the auxiliary coefficients as
S00 =
1− I00i − ω2di
2ω2I20i
+ τ−1
1− I00e − ω2de
2ω2I20e
,
S∗00 = I00∗i −ω2di2ω2
I20i −I00∗e −
ω2de2ω2
I20∗e
,
S′00 =1− I00i
+ τ−1
1− I00e
,
S′∗00 = I00∗i − I00∗e ,
S01 = Γ0i − 1 +ω2di2ω2
I20i + τ−1Γ0e − 1 +
ω2de2ω2
I20e
,
S03 =I10i − I10e
,
S∗03 =
I10∗i +
ω∗eω∗i
I10∗e
.
28
2.2.3 Dispersion relation
Using above Eq. (2.19) in Eq. (2.20), we obtain
φ1s = iωdiω
S03S00
S′00S00 +ω2∗iω2
S′∗00S∗00φ0, (2.21)
and substituting Eq. (2.21) in Eq. (2.20), we get
φ1c = iωdiω
ω∗iω
S03S∗00
S′00S00 +ω2∗iω2
S′∗00S∗00φ0. (2.22)
Solving Eq. (2.18) with the help of Eqs. (2.21, 2.22), the dispersion relation becomes
S01 +ω2di2ω2
S03
S03S00 −
ω2∗iω2
S∗03S∗00
S′00S00 +ω2∗iω2
S′∗00S∗00= 0 (2.23)
The above dispersion relation is the full kinetic dispersion relation of electrostatic GAMs which
includes the effects of both the temperature and density inhomgeneities, the Landau and toroidal
resonances.
2.2.4 Simplified Dispersion Relations
Assuming that ions are in the fluid regime (ω >> vti/qR0), and the electrons are in the
adiabatic regime (ω << vte/qR0), the general integrals in Appendix A may be expanded as
given in Appendixes. (B, C) and resultantly we obtain
ω2 − v2tiR20
7
4+
τ + (1 + ηi)ω2∗/ω
2
τ2 − (ω2∗/ω2)
= 0. (2.24)
2.3 Two Fluid MHD Model
In the study of GAM, it is reasonable to assume that ions are in the fluid regime, ω >>
vti/qR0, and the electrons are in the adiabatic regime, ω << vte/qR0. In this context, electron
temperature fluctuations can be neglected in the lowest order and the electron has a Boltzmann
response as consequence. On the other hand, for ions, parallel viscosity, π =3π(bb− I/3)/2,
29
must be consider to account for pressure anisotropy (π = 2(p − p⊥)/3) effects.
In this chapter, we use ideal MHD equations for ion and electrons in the form:
∂n
∂t+∇. (nVE) = 0, (2.25)
3
2
dp
dt+ v.∇p
+
5
2p∇.v+∇.q = 0, (2.26)
dπdt
+ p
−2v.∇ lnB − 2
3∇.v
+
2
5
−2q.∇ lnB − 2
3∇.q
= 0, (2.27)
mndv
dt+∇p+∇.π − eαn(E+ v×B) = 0, (2.28)
We note that Eqs. (2.25, 2.26, 2.28), representing conservation of mass, energy and momentum
respectively, have been widely used in both one and two fluid models. Eq. (2.27) is the parallel
component of Grad type equation for the viscosity tensor, π, that was derived for general
curvilinear magnetic field in Ref. [32]. Parallel velocity, v, which is responsible for O(q2)
corrections, is neglected in our analysis since GAM is mostly important at the edge of the
plasma column.
Taking the cross product of Eq. (2.28) withB/B = b, one finds the equation for ion/electron
currents
J = JI + Jp + Jπ + J + eαnVE , (2.29)
where
JI =eαn
ωcα
b×dv
dt
,Jp =
b×∇p
B,Jπ =
b×∇.π
B, J = Jb,
VE =b×∇φ
Band E = −∇φ−
∂A∂θ
b.
Eqs. (2.25- 2.29) are closed with quasi-neutrality condition in the forms
e(ni − ne) = 0, ∇.J = 0. (2.30)
Here, the subscript α = e, i, standing for ions and electrons species, were omitted in n, v, p, q,
30
m and π for simplicity of notation. As in standard ideal MHD models, frozen field condition
E+ v×B = 0, (2.31)
from Eq. (2.28), is considered to find the velocity in a first approximation [27]. The velocity,
VE , is then used in the continuity equation for ions
∂ni∂t
+VE .∇n0 + n0∇.VE = 0, (2.32)
to find the first side-bands components of ion density, ni±1. Here the second term is responsible
for the density drift effects. The following perturbed quantities, φ, n, p, π, are assumed to be
in the form
X = X0 +X1 exp(iθ) +X−1 exp(−iθ), (2.33)
since for the study of GAM only poloidal modes M = 0,±1 and the toroidal mode N = 0 are
important. Then, by using (2.33) into (2.32) we obtain
ni±1 =
± i
2
ωdiω
φ0 ∓ω∗ω
φ±1
, (2.34)
where ωdi = krρivti/R0 is the magnetic drift frequency, ω∗ = ω∗i is the density drift frequency
defined by ω∗i = Tin′0/ (reB0n0) = ρivti/ (2rLN) and LN = (d lnn0/dr)
−1 is the characteristic
density scale length. To solve (2.32) we use∇.VE = −2VE.∇ lnB and we consider high aspect
ratio approximation, ǫ = r/R0 << 1, and circular magnetic surfaces tokamaks, which gives B
≈ B0(1− ǫ cos θ) and consequently ∇ lnB = (− cos θer + sin θeθ)/R0.
For electrons, due to its small mass, the parallel component of the momentum equation can
be approximated by
Te∇ne − en0∇φ = 0, (2.35)
which leads to the Boltzmann response as solution for the first side-bands,
ne±1 =en0Te
φ±1. (2.36)
By substituting (2.34) and (2.36) in the quasi-neutrality condition, e(ni − ne) = 0, we obtain
31
the relation between the potentials,
φ±1 = ± i
2
ωdi/ω
τ ± ω∗/ωφ0, (2.37)
in according with Refs. [39], [49] (Eqs. (8) and (3.8), respectively) using kinetic model. Here
the ratio of temperatures is τ = Ti/Te.
The reduced energy equation for ions is obtained in the form
3
2
dpidt
+VE .∇p0i
+
5
2p0i∇.VE = 0. (2.38)
It is solved for the first side-bands of ion pressure,
pi±1 =
±5
3
i
2
ωdiω
φ0 ∓ (1 + ηi)ω∗ωφ±1
en0, (2.39)
where ηi = Ln/LT and d lnT/dr is the characteristic ion temperature scale length. It is worth
noting here that, in general, the heat flux (q), nor the diamagnetic drift, are not divergence
free in toroidal geometry. It can be shown however, that these effects do not contribute to the
GAM dispersion in the leading order.
In case of electrons, since fluctuations of temperature are small in the adiabatic regime,
electron pressure can be written as
pe±1 = ±φ±1en0. (2.40)
From the reduced ion parallel viscosity equation [27]
dπidt
− 2
3poiVE .∇ lnB = 0, (2.41)
in which heat flux contributions can be neglected, we obtain the first side-bands of parallel
viscosity
πi±1 = ± i
6
ωdiω
φ0en0, (2.42)
where it can be noted that this equation is unchanged by diamagnetic effects.
Considering that electrons are isotropic and so πe = 0, it is convenient to compute the sin θ
32
component of the combination p+ π from both ions and electrons contribution
(p+ π/4)s = −ωdiω
7
4+
τ + (1 + ηi)ω2∗/ω
2
τ2 − (ω2∗/ω2)
φ0en0. (2.43)
This will be used in the current conservation equation averaged over the magnetic surfaces
∇.J = 0, (2.44)
where J is the sum of the ion and electron currents and the average over the magnetic surfaces,,is used to eliminate the parallel current contributions, since
∇J
= 0. In Eq. (2.44), since
ωci << ωce and the ion and electron currents due to VE cancel each other, we need to consider
ion inertial current, the total (ion and electron) diamagnetic current, and the viscosity current
for ions.
The divergence of these currents averaged over the magnetic surfaces are calculated in the
leading order as follows:
∇.JI ≈en0ωcib×dVE
dt
≈iωen0ωci
∇.∇⊥φ
B
≈ −4π2rR0iω
en0ωci
k2r φ0B0
, (2.45)
∇.Jp +∇.Jπ ≈ −2Jp.∇ lnB + Jπ.∇ lnB ≈−2ikrp sin θ
R0B0−
ikrπ sin θ
2R0B0
,
≈ −4π2rR0ikr (p+ π/4)s
R0B0. (2.46)
Finally, the dispersion relation can then be written in the form
ω2 − v2tiR20
7
4+
τ + (1 + ηi)ω2∗/ω
2
τ2 − (ω2∗/ω2)
= 0. (2.47)
This equation has following solutions:
2Ω2± = ω2gam + ω2∗e ±
ω2gam + ω2∗e2
+ (4ηi − 3)ω2∗ev2ti/R
20. (2.48)
where ω2∗i = −ω2∗e/τ is the electron drift frequency and ω2gam = (7/4 + 1/τ)v2ti/R20 is the GAM
frequency in the absence of drift effects.
It can be seen from this equation, that diamagnetic effects due to temperature and density
33
gradients modify the GAM frequency and create a new mode which becomes unstable for larger
values of ηi > 3/4. The new mode occurs as a result of coupling to the degenerate (in absence
of drift effects) zero frequency mode normally associated with zonal flow. For weak gradient,
ω∗e << ωgam(ρi/Ln << r/R0), the two branches of the solution can be approximated by
Ω2+ = ω2gam +1 + ηi + 1/τ
7/4 + 1/τω2∗e and Ω2− =
3/4− ηi7/4 + 1/τ
ω2∗e (2.49)
The Ω+ branch is related to the GAM mode, and the Ω− branch is related to the ZF. The
zonal flow branch becomes unstable for large temperature gradient, ηi > 3/4.
34
Chapter 3
Electromagnetic Effects on Geodesic
Acoustic Modes
By using the full electromagnetic drift kinetic equations for electrons and ions, the general dis-
persion relation for geodesic acoustic modes (GAMs) is derived incorporating the electromag-
netic effects. It is shown that m=1 harmonic of the GAM mode has a finite electromagnetic
component. The electromagnetic corrections appear for finite values of the radial wave numbers
and modify the GAM frequency. The effects of plasma pressure βe, the safety factor q and the
temperature ratio τ on GAM dispersion are analyzed.
3.1 Introduction
Geodesic Acoustic Modes (GAMs) are linear eigenmodes of poloidal plasma rotation supported
by plasma compressibility in toroidal geometry and were first predicted by Winsor et al., [16].
They are closely related to Beta-induced Alfvén eigenmodes (BAEs) [17, 18, 44, 55, 56]. Geo-
desic Acoustic Modes have attracted much interest both experimentally and theoretically due
to their possible role in drift-wave turbulence and plasma transport. The coupling of GAMs,
zonal flows, and small scale drift-wave fluctuations via toroidal effects and nonlinear Reynolds
stress has been noted in a number of simulations [57]-[59]. The fluctuations with GAM signa-
tures and in the GAM frequency range have been observed in numerous devices [51], [61]-[63].
It has been suggested that GAMs can be used also for magnetohydrodynamic spectroscopy and
35
diagnostics of the current profile in burning plasmas [64, 65].
The kinetic aspects of GAM modes have been investigated theoretically by several authors
[68]-[70]. One of the important issue is the mode dispersion which determines the radial prop-
agation and radial structure of GAMs [68, 71, 72, 62]. It has been shown that the second
sideband harmonics of GAM (which are generally electromagnetic) are important for form-
ing the eigenmode structure in MHD models [56, 66, 67]. Most of kinetic studies so far have
considered electrostatic approximation and neglected the magnetic component of the GAM.
Electromagnetic effects were considered in Refs. [69, 73], however it was concluded [69] that
the magnetic component of the first side bands of GAM is identically zero and therefore only
the second order sidebands can be electromagnetic. The electromagnetic effects of the second
sidebands on the GAM frequency and damping rates were kinetically analyzed in Ref. [73]. It
was noted however [20] that global GAMs with large radial extent should have substantial elec-
tromagnetic component. It is shown in the present paper that for higher plasma pressure, the
first sideband of the perturbed magnetic potential can be of the same order as the electrostatic
sideband and the corresponding expression for the magnetic component is derived.
For the analytical derivation of the dispersion relation, we assume that all perturbations
vary as X = X0 exp+ X±1 exp (±iθ) + X±2 exp (±2iθ) , where X0 and X±1, X±2 represent the
principal and sideband components. Respectively, the wave vectors for the sidebands are k± =
±1/qR, k2± = ±2/qR and the principal harmonic has k0 = (m− nq) /qR =0. In this paper,
we solve the full electromagnetic drift kinetic equations for electrons and ions and obtain the
general dispersion relation for the principal m = n = 0 mode and m = 1 sidebands only. The
general dispersion relation incorporates the effects of the finite value of the parallel wave vector
k± (both for electrons and ions) and radial dispersion due to electromagnetic effects . We
also obtain the simplified dispersion relation by assuming that the electrons are in adiabatic
regime ω < vTe/qR whereas the ions are in the fluid regime ω > vTi/qR. The main finding
of the present work is that the parallel current in the first order is finite resulting in magnetic
nature of the sidebands controlled by the radial mode width parameter K⊥ = ckrλDe/ωqR.
The electromagnetic corrections contribute to the mode dispersion and have to be included in
the dispersive models of GAMs that are required for the determination of the GAM eigenmode
properties. We also show the route for the self-consistent expansion including the second order
36
harmonics. This is further discussed in the Appendix E.
In the followings, we solve the drift kinetic equation and obtain the general kinetic dispersion
relation of GAMs and then by simplifying the electron and ion responses, the dispersion relation
is obtained in the limit when the electrons are in the adiabatic regime (ω < vTe/qR) and the
ions are in the fluid regime(ω > vTi/qR) . At the end, the summary of the results is given.
3.2 Drift Kinetic Equation and Full Kinetic Dispersion Relation
We use the drift kinetic equation in the form
fα = −eαφ
TαFMα + gα, (3.1)
where the g distribution function satisfies the equation
ω − ωdα − kv
gα = ωJ20 (zα) (φ−
v
cA)
eαFMα
Tα. (3.2)
Here
ωdα = −v2⊥/2 + v2
ωcα
krR
sin θ = − ωdα sin θ, ωdα = ωdαv2⊥/2 + v2
v2tα,
ωdα =krv
2tα
Rωcα, ωcα =
eαB0mαc
, v2tα =2Tαmα
and zα =krv⊥ωcα
.
Separating the principal and the oscillating components of Eq. (3.2), we obtain
ωg0α − ωdαgα = ω(φ0 −vcA0)J
20 (zα)
eαFMα
Tα,
(3.3)
ωgα − ωdαg0α − (ωdαgα − ωdαgα)− kvgα = ω(φ−vcA)J20 (zα)
eαFMα
Tα.
(3.4)
The notation ... represents the average in θ. The magnetic vector potential A describes the
electromagnetic effects and leads to the higher order corrections to the density perturbation
and the parallel current.
37
Using the perturbation of the form
X = X0 +X1eiθ +X−1e
−iθ = X0 +X1c cos θ + iX1s sin θ,
Eqs. (3.3,3.4) are simplified to the form
ωg0 +iωdα2
g1s = ω(φ0 −v
cA0)J
20 (zα)
eαFMα
Tα,
(3.5)
ωg1c −vqR
g1s = ωφ1c −
vcA1c
J20 (zα)
eαFMα
Tα,
(3.6)
−iωdαg0 −vqR
(g1 + g−1) + ω (g1 − g−1) = ωφ1s −
v
cA1s
J20 (zα)
eαFMα
Tα,
(3.7)
where we have introduced the notations X1c,s = X1 ±X−1.
Solving the Eqs. (3.5-3.7) simultaneously with k0 = 0 and correspondingly A0 = 0, we get
the principal and side band components as
g0 = −J20 (zα)
eαFMα
Tα
W
iωdαω
2
φ1s −
vcA1s
+
iωdαv2qR
φ1c −
vcA1c
−ω2 −
v2q2R2
φ0
,
(3.8)
g1c = −J20 (zα)
eαFMα
Tα
W
−ω2 − ω2
dα
2
φ1c −
vcA1c−
vω
qR
φ1s −
vcA1s
−iωdαv
qRφ0
,
(3.9)
g1s = −J20 (zα)
eαFMα
Tα
W
−ω2
φ1s −
vcA1s
−
vω
qR
φ1c −
vcA1c
− iωωdαφ0
,
(3.10)
where
W = ω2 −v2
q2R2− ω2dα
2.
38
Using the Eqs. (3.8-3.10) in Eq. (3.1), the perturbed distributions simplify to
f0 = −eαFMα
Tα
φ0 +
J20 (zα)
W
iωdαω
2
φ1s −
vcA1s
+iωdαv2qR
φ1c −
vcA1c
−ω2 −
v2q2R2
φ0
,
(3.11)
f1c = −eαFMα
Tα
φ1c +
J20 (zα)
W
−ω2 − ω2
dα
2
φ1c −
v
cA1c
−vω
qR
φ1s −
v
cA1s
−iωdαvqR
φ0
,
(3.12)
f1s = −eαFMα
Tα
φ1s +
J20 (zα)
W
−ω2φ1s −
vcA1s
−vω
qR
φ1c −
vcA1c
− iωωdαφ0
.
(3.13)
3.2.1 Density Perturbation
Now we calculate the density perturbation
n =
fd3v. (3.14)
Using Eq. (3.11-3.13) in Eq. (3.14), we obtain the following expressions for various compo-
nents of density perturbation
n0α = −eαN0Tα
1− Γ0α −
ω2dα2ω2
I20α
φ0 + i
ωdα2ω
I10α φ1s − iωdα2ω
I11αs2α
ωqR
cA1c
, (3.15)
n1cα = −eαN0Tα
(1− Γ0α)φ1c −
I01αs2α
φ1c −
ωqR
cA1s
, (3.16)
n1sα = −eαN0Tα
1− I00α
φ1s +
I01αs2α
ωqR
cA1c − i
ωdαω
I10α φ0
. (3.17)
The definitions of various integrals are given in Appendix A, and sα = ωqR/vtα. The
39
quasi-neutrality equation together with Eqs. (3.15-3.17) gives
S01φ0 + iωdi2ω
S02ωqR
cA1c −
i
2
ωdiω
S03φ1s = 0, (3.18)
φ1c −
ωqR
cA1s
= −
(Γ0i − 1) + τ−1 (Γ0e − 1)
D1
φ1c, (3.19)
φ1s −
ωqR
cA1c
=
S1D1
φ1s − iωdiω
S03D1
φ0, (3.20)
where we have defined the auxiliary coefficients as follows
S01 = Γ0i − 1 +ω2di2ω2
I20i + τ−1Γ0e − 1 +
ω2de2ω2
I20e
,
S02 =I11is2i− I11e
s2e,
S03 =I10i − I10e
,
S1 = 1− I00i +I01is2i
+ τ−11− I00e +
I01es2e
,
D1 =I01is2i
+ τ−1I01es2e
,
and the temperature ratio as τ = Te/Ti.
3.2.2 Current Density
The perturbed distribution function in Eqs. (3.12,3.13) can be used to calculate the parallel
current perturbation
j = eα
fvd
3v,
so that we obtain
j1cα = −e2αN0αTα
ωqR
Γ0α
s2α+
I02αs4α
ωqR
cA1c −
I01αs2α
φ1s − iωdαω
I11αs2α
φ0
, (3.21)
j1sα =e2αN0αTα
ωqRI01
s2α
φ1c −
ωqR
cA1s
. (3.22)
40
Total parallel current of Eq. (3.21) as a sum of electron and ion contributions becomes
j1c = j1ce + j1ci =e2N0Ti
ωqR
S2
φ1s −
ωqR
cA1c
− [S2 −D1]φ1s + i
ωdiω
S02φ0
, (3.23)
where
S2 =
Γ0i
s2i+
I02is4i
+ τ−1
Γ0e
s2e+
I02es4e
.
Using Eq. (3.20) in Eq. (3.23), the total parallel current can be reduced to the form
j1c =e2N0Ti
ωqR
S1S2D1
− S2 +D1
φ1s − i
ωdiω
S03S2D1
− S02
φ0
. (3.24)
Using the Ampere’s law k2rA1c = 4πc j1c gives the expression for the perturbed vector potential
A1c =τc
K2⊥ωqR
S1S2D1
− S2 +D1
φ1s − i
ωdiω
S03S2D1
− S02
φ0
. (3.25)
The parameter K2⊥, which is responsible for the electromagnetic natures of the GAM [20], is
defined as
K2⊥ =
c2k2rλ2De
ω2q2R2.
Note that for GAM range frequencies ω2 ≃ v2Ti/R2, the K2
⊥ parameter becomes K2⊥ ≃
τk2rc2/q2ω2pi
≃ τk2rρ
2i /βiq
2. Since we are interested only in GAM mode, only the cos θ
parity current is shown here. The sin θ parity can be calculated similarly.
3.2.3 Full Kinetic Dispersion relation
The Eqs. (3.18, 3.20, 3.25) are used to derive the general dispersion relation. After putting Eq.
(3.25) in Eqs. (3.18, 3.20), we get
S01 +
ω2di2ω2
τS202K2⊥
S03S02
S2D1
− 1
φ0 −
i
2
ωdiω
S03 −
τ
K2⊥
S02
S1S2D1
− S2 +D1
φ1s = 0,
(3.26)
41
φ1s = −iωdiω
S03D1
+τS02K2⊥
S03S02
S2D1
− 1
1− τ
K2⊥
S1S2D1
− S2 +D1
− S1
D1
φ0. (3.27)
Solving the above two equations yields
S01 +ω2di2ω2
τ
K2⊥
S202
S03S02
S2D1
− 1
− ω2di2ω2
S202D1
S03S02
+τD1
K2⊥
S03S2S02D1
− 1
1 +
S03S02
− 1 +S1D1
1− τ
K2⊥
S1S2D1
− S2 +D1
− S1
D1
= 0.
(3.28)
The above equation represents the general dispersion relation of GAMs with full kinetic
response of electron and ions including the electromagnetic corrections. The latter depend on
the radial mode width characterized by the K⊥ parameter. The limit of large (small) value
of the parameter K2⊥ > 1
K2⊥ < 1
describes the electrostatic (electromagnetic) behavior of
the sidebands. This equation can be used to investigate the role of electromagnetic effects on
the mode dispersion and mode damping. This equation includes both Landau ω ≃ v/qR and
toroidal ω ≃ ωdα resonances.
3.3 Simplified Dispersion Relation
In this section we derive the simplified dispersion equation by neglecting the resonant effects
i.e., assuming that the electrons are in the adiabatic regime (se < 1) and ions are in the fluids
regime si > 1. We also assume the small Larmor radius limit so that the toroidal resonances can
be neglected as well: ω2de < ω2 and ω2di < ω2. In this limit, the expanded form of the integrals
can be used as given in Appendices (B, C). The simplified dispersion equation can be directly
derived from the full expression (3.28) by using Appendix D, but we give here all simplified
expressions for plasma density and potentials to highlight the nature of the magnetic correction
and estimate its amplitude. In this section we only show the components n1se, φ1s and A1c
42
relevant to GAM polarization [31].
3.3.1 Electron Response
The simplified electron response is obtained from Eqs. (3.15-3.17, 3.21), by using the expansions
given in Appendix B, as
n0e = − i
2
en0Te
ωdeω
ωqR
cA1c, (3.29)
n1ce =en0Te
φ1c −
ωqR
cA1s
, (3.30)
n1se =en0Te
φ1s −
ωqR
cA1c
, (3.31)
j1ce =e2n0Te
ω2q2R2
cA1c − ωqRφ1s + iωdeqRφ0
. (3.32)
These expressions can also be obtained from fluid equations [31]. Expressions (3.30,3.31) follow
from the electron momentum balance equation in the adiabatic limit ω < vTe/qR. The equa-
tions (3.29) and (3.32) can be obtained respectively from the m = 0 and m = 1 components of
the electron continuity equation and using Eqs. (3.30, 3.31).
3.3.2 The Ion Response
Similarly, using the Appendix C, we may simplify the ion equations (3.15-3.17, 3.21) by using
the expansions ω > vTi/qR and ωDi < ω:
n0i =eN0Ti
Γ0i − 1 +
1
2
K2 +K2
φ0 −
i
2
K1 +K1
φ1s +
i
2K1
ωqR
cA1c
,
(3.33)
n1ci = −eN0Ti
φ1c (1− Γ0i)−
φ1c −
ωqR
cA1s
K
,
(3.34)
n1si =eiN0Ti
Γ0i − 1 +K +
1
2K2 +
1
2K2
φ1s −K
ωqR
cA1c + i
K1 +K1
φ0
,
(3.35)
j1ci =e2N0Ti
ωqR
K
φ1s −
ωqR
cA1c
+ iK1φ0
.
(3.36)
43
The terms with Γ0i − 1 are responsible for inertial (polarization) corrections, the terms
with K are due to parallel ion motion (ion sound dynamics), the terms with K1 are due to
toroidal compressibility, and K1 is the combination of the toroidal compressibility and ion
sound dynamics. Note that the full parallel electric field E = −iφ1c −
ωqR
cA1s
/qR, which
includes the electromagnetic part is used to calculate the parallel ion response.
3.3.3 The Dispersion Relation and Magnetic Perturbation Effect
The quasi-neutrality conditions for principal and oscillating components take the form
0 =
Γ0i − 1 +
1
2K2 +
1
2K2
φ0 −
i
2
K1 +K1
φ1s −
ωqR
cA1c
, (3.37)
φ1c −
ωqR
cA1s
=
(Γ0i − 1)τ−1 −K
φ1c, (3.38)
φ1s −
ωqR
cA1c
=
Γ0i − 1 +1
2K2 +
1
2K2
τ−1 −K
φ1s + i
K1 +K1
τ−1 −K
φ0. (3.39)
Total parallel current
j1c = −e2N0Ti
ωqR
φ1s −
ωqR
cA1c
τ−1 −K
− i
K1 +K1
φ0
. (3.40)
Using Eq. (3.39) in above equation, we obtain the parallel current in terms of the cos θ harmonic
of the electrostatic potential
j1c = −ωqRe2N0Ti
(Γ0i − 1 +1
2
K2 +K2
)φ1s. (3.41)
With the Ampere’s law one obtains general expression for the perturbation of the magnetic
vector potential
A1c = − τc
K2⊥ωqR
Γ0i − 1 +
1
2
K2 +K2
φ1s. (3.42)
The expression (3.41) creates the impression that the parallel current is the finite Larmor radius
effects, as it was assumed in Refs. [69, 73]. In fact, the GAM involves the balance of the ion
polarization current and the divergence of the diamagnetic current, so it is in the same order
44
as the magnetic perturbation given by Eq. (3.42). It is easy to see that radial wave vector kr
cancels out from Eq. (3.42).
It is instructive at this point to compare our expressions with those in Refs. [69, 73]. Our
expression for the parallel current j1c in Eq. (3.41) is identical to the fluid limit of the expression
(5) of Ref. [69]. We believe that the identical expression was used in Refs. [69, 73]. The most
important difference is in the expression for the sideband of the ion density ( n1si) in Eq. (3.35).
The last term in this expression is identical to the ones used in Eq. (4b), Ref. [69] and Eq.
(11), Ref. [73]. The terms with φ1s and A1c in our Eq. (3.35) are absent in Refs. [69, 73].
These terms result in additional contribution with φ1s on the right hand side of Eq. (3.39).
This is exactly this term (absent in Ref. [73]) which makes the parallel current finite in our Eq.
(??). This current was obtained to be zero in Refs. [73].
The combination in square brackets in (3.42) is the GAM dispersion relation in the lowest
order excluding the electron contribution to the diamagnetic current.
Including the perturbed magnetic potential from Eq. (3.42), the Eqs.(3.37, 3.39) simplify
to
Γ0i − 1 +
1
2K2 +
1
2K2
φ0 −
i
2
K1 +K1
1 +
Γ0i − 1 +1
2
K2 +K2
τ−1K2⊥
φ1s = 0,
(3.43)
φ1s
1 +
Γ0i − 1 +
1
2
K2 +K2
1
τ−1K2⊥
+1
K − τ−1
+ i
K1 +K1
K − τ−1
φ0 = 0,
(3.44)
leading to the general dispersion relation
45
(Γ0i − 1 +1
2K2 +
1
2K2)
+1
2
K1 +K1
2τ−1 −K
1 +Γ0i − 1 +
1
2
K2 +K2
τ−1 −K −Γ0i − 1 +
1
2
K2 +K2
K2⊥ − 1 + τK
K2⊥
= 0.
(3.45)
The factor τ−1−K = 0 is responsible for the ion sound branch: ω2 = Te/miq
2R2. The
seconds term in bracket is generally higher order correction. In the lowest order, one obtains
the dispersion relation reduces to
(Γ0i − 1 +1
2K2 +
1
2K2) +
1
2
K1 +K1
2τ−1 −K
= 0. (3.46)
Using the expansionτ−1 −K
= τ−1
1 + τK
one obtains from (3.46) the equation that
was obtained earlier for GAM [35, 45]:
ω2 =v2TiR2
7
4+ τ
+
v2TiR2
v2Tiω2q2R2
23
8+ 2τ +
τ2
2
. (3.47)
The magnitude of the magnetic perturbation in (3.42) can be estimated by using the lowest
order solution ω2 = v2Ti/R2 (7/4 + τ). Then
Γ0i − 1 +1
2
K2 +K2
≃ −τK2
1/2 , (3.48)
so that
A1c ≃ βeq2 c
ωqRφ1s, (3.49)
where βe = 8πn0Te/B2. Note that the amplitude of the first side-band harmonic φ1s is related
to the m = n = 0 harmonic by the relation
φ1s ≃ 2τkrρiφ0. (3.50)
46
Plasma parameters scalings from the order of magnitude estimates (3.49) and (3.50) can be
used to identify the signatures of magnetic fluctuations in experiments.
The effects of magnetic component in Eq. (3.45) is controlled by the amplitude of the radial
mode width parameter [20] K2⊥. The electrostatic regime corresponds to the case K2
⊥ ≫ 1. In
this limit one has from Eq. (3.45)
(Γ0i − 1 +1
2K2 +
1
2K2) +
1
2
K1 +K1
2
τ−1 −K −Γ0i − 1 +
1
2
K2 +K2
= 0.
(3.51)
One can see that this limit can be obtained by setting the magnetic vector potential A1c = 0
in Eqs. (3.29-3.36). In the electromagnetic limit, K2⊥ ≪ 1, one has from Eq. (3.45)
(Γ0i − 1 +1
2K2 +
1
2K2)
+1
2
K1 +K1
2τ−1 −K
1 +
Γ0i − 1 +1
2
K2 +K2
τ−1 −K −Γ0i − 1 +
1
2
K2 +K2
/K2
⊥
= 0,
(3.52)
or
(Γ0i − 1 +1
2K2 +
1
2K2) +
1
2
K1 +K1
2
τ−1
1− K2
1τ2
2 (1 + βeq2)
= 0.
(3.53)
General structure of the roots of Eq. (3.45) is shown in Fig. (3.1) for different values of plasma
pressure along with the lowest order standard dispersion relation of GAMs
ω2 =
v2tiR2
7
4+ τ
and ion sound mode
ω2 =
v2tiτ
2q2R2
. The upper roots above the GAMs correspond to the de-
nominator of the dispersive part in expression (3.45) and are related to the Alfvén sideband
continuum ω = vA/qR branches. The lower frequency ion sound modes (below the GAM
frequency) have also been found earlier in kinetic analysis [74, 19]. The standard and lower
47
frequency ion sound modes and higher frequency modes (above GAM) will not be considered
here. So only the GAMs root is shown in Figs.(3.2-3.8). Figs.(3.2, 3.3) depict the normalized
frequencyωR
vtias function of krρi to highlight the dispersion effect on GAMs. Note that the
mode dispersion due to magnetic effect is negative. In Figs.(3.4-3.6) , we show the effect of
plasma pressure βe on GAMs for various values of plasma parameters and value of krρi. It
follows that finite plasma pressure reduces the mode frequency as shown in Figs.(3.4-3.6). The
role of the magnetic corrections on the mode dispersion is shown in Figs. (3.7) and (3.8). Here
we show the simplified leading order dispersion relation for GAMs from Eq. (3.47) in compar-
ison with the solution from full Eq.(3.45) as a function of temperature ratio τ and the safety
factor q respectively. Figs. (3.7) and (3.8) show that the mode frequency decreases with an
increase of krρi in consistence with negative dispersion due to magnetic effects.
Figure 3-1: Multiple roots of the dispersion relation (3.45) i.e., Normalized frequencyωR
vtivsas functions of krρi at different values of plasma pressure: βe =0.05 — solid line, βe = 0.07—dotted line, βe = 0.1 — dashed line, βe = 0.2 — dot-dashed line; q = 1.8 and τ = 1.6.
48
Figure 3-2: Normalized FrequencyωR
vtivs krρi for GAM branch at fixed q = 1.8 and βe = 0.2
for different values of the temperature ratio: τ = 1.6 — solid line, 1.7 — dotted line, 1.8 — dashedline, 1.9 — dot-dashed line.
Figure 3-3: Normalized frequencyωR
vtivs krρi for GAM branch at fixed τ = 1.3 and βe = 0.1
for different values of safety factor: q = 1.8 — solid line, 2.5 — dotted line, 3.5 — dashed line, 4.5— dot-dashed line.
49
Figure 3-4: Normalized frequencyωR
vtivs βe for GAM branch at fixed q = 2.5 and τ = 1.3 for
different values of normalized radial wave number: krρi = 0.1 — solid line, 0.2 — dotted line, 0.3— dashed line, 0.4 — dot-dashed line.
Figure 3-5: Normalized frequencyωR
vtivs βe for GAM branch at τ = 1.8 and krρi = 0.4 with
variation in safety factor: q = 2.0 — solid line, 2.5 — dotted line, 3.0 — dashed line, 3.5 —dot-dashed line.
50
Figure 3-6: Normalized frequencyωR
vtivs βe for GAM branch at fixed value of q = 2.5 and
krρi = 0.3 for different values of temperature ratio: τ = 1.0 — solid line, 1.3 — dotted line, 1.6 —dashed line, 1.9 — dot-dashed line.
Figure 3-7: Normalized frequencyωR
vtivs τ for GAM branch at fixed q = 3.0 as a function of
τ : a) βe = 0.1, Eq.(3.47) — solid line, krρi = 0.05 — double dotted, krρi = 0.1 — dotted line,krρi = 0.2 — dashed line, krρi = 0.4 — dot-dashed line; b) krρi = 0.1, Eq. (3.47) — solid line,βe = 0.1 — double dotted, βe = 0.2 — dotted line, βe = 0.3 — dashed line, βe = 0.4 — dot-dashedline.
51
Figure 3-8: Normalized frequencyωR
vtivs q for GAM branch at fixed τ = 1.6 as a function
of q, a) βe = 0.03, Eq. (3.47) — solid line, krρi = 0.2 — double dotted, krρi = 0.3 — dottedline, krρi = 0.4 — dashed line, krρi = 0.5 — dot-dashed line; b) krρi = 0.1, Eq. (3.47) — solidline, βe = 0.05 — double dotted, βe = 0.1 — dotted line, βe = 0.2 — dashed line, βe = 0.4 —dot-dashed line.
52
Chapter 4
Stability Analysis of
Self-Gravitational Electrostatic Drift
waves for a Streaming Non-Uniform
Quantum Dusty Magnetoplasma
Using the quantum hydrodynamical model of plasmas, the stability analysis of self-gravitational
electrostatic drift waves for a streaming non-uniform quantum dusty magnetoplasma is pre-
sented. For two different frequency domains i.e., Ω0d << ω < Ω0i (unmagnetized dust)
and ω << Ω0d < Ω0i (magnetized dust), we simplify the general dispersion relation for self-
gravitational electrostatic drift waves which incorporates the effects of density inhomogeneity
∇n0α, streaming velocity v0α due to magnetic field inhomogeneity ∇B0, Bohm potential and
the Fermi degenerate pressure. For both frequency domains, the effect of density inhomogene-
ity gives rise to real oscillations while the ions streaming velocity v0i as well as the effective
electron quantum velocity v′Fe make these oscillations propagate perpendicular to the ambient
magnetic field. This oscillatory behavior of self-gravitational drift waves increases with increase
in inhomogeneities and quantum effects while it decreases with increase in the gravitational po-
tential. However only for the unmagnetized case, the drift waves may become unstable under
appropriate conditions giving rise to Jeans instability. The modified threshold condition is also
53
determined for instability by using the intersection method for solving the cubic equation. We
note that the inhomogeneity in magnetic field (equilibrium density) through streaming veloc-
ity (diamagnetic drift velocity) suppress the Jeans instability depending upon the characteristic
scale length of these inhomogeneities. On the other hand, the dust-lower-hybrid wave and the
quantum mechanical effects of electrons tend to reduce the growth rate as expected. A number
of special cases are also discussed.
4.1 Introduction
The problem of gravitational collapse of astrophysical nebulae for the formation of stars and
galaxies has been a great challenge in astrophysics and cosmology. James Jeans [79] first showed
how a neutral fluid in a nebula containing micron-sized dust grains may become unstable due to
its own self-gravity. This is the main mechanism for the large scale nebulae in the Universe to
collapse to stars, galaxies, etc. or other structures and their evolution. It is also known that the
Jeans instability is a relatively faster process, whereas the formation of heavenly objects takes
place in billions of years [80]-[85]. Obviously, there should be a number of hindering effects on
Jeans instability, which may explain the real phenomena behind the gravitational collapse.
It is believed that due to the presence of all-pervading ultra-violet photons, plasma currents
or for some heavenly occurrences, the micron-sized dust grains of the collapsing systems can
acquire electric charges and thus a self-gravitating dusty plasma under extreme conditions may
be formed [86]-[88]. These plasmas may contain static and inhomogeneous ambient magnetic
field, nonuniform densities or even quantum effects under extreme conditions.
In recent years, there has been a growing interest in quantum plasmas because of their
importance in microelectronics and electronic devices with nano-electronic components [89, 90],
dense astrophysical systems [91]-[93], and in laser-produced plasmas [94]-[97]. When a plasma
is cooled to an extremely low temperature, the de Broglie wavelengths of the plasma particles
could be comparable to the scale lengths, such as Debye length or Larmor radius, etc. in the
system. In such plasmas, the ultracold dense plasma would behave as a Fermi gas and the
quantum mechanical effects might play a vital role in the behavior of the charge carriers of
these plasmas under the extreme conditions.
54
Extensive studies have been done over the years by taking into account a wide variety of
effects so as to study Jeans instability. Shukla and Stenflo [98] examined the influence and
the range of validity of quantum effects on Jeans instabilities of homogeneous and unmagne-
tized self-gravitating astrophysical quantum dusty plasma systems where the electromagnetic
and gravitational forces on plasma charge carriers become comparable. Ren et al.[107, 108]
investigated the Jeans instability in a dense quantum plasma in the presence of two dimen-
sional magnetic fields and the resistive effects with or without Hall current effects. Recently,
Prajapati and Chhajlani [109] discussed the contributions of Hall current and viscosity of the
medium to Jeans instability and its importance to astrophysical plasmas. The inhomogeneous
ambient magnetic field, the nonuniform plasma density, and the quantum effect might play an
important role in reducing the growth rate of the Jeans instability of the real physical plasmas.
In an earlier paper [28], we also examined Jeans instability in a homogeneous dusty plasma
in the presence of an ambient magnetic field with quantum effect arising through the Bohm
potential only.
In this chapter, we present a detailed investigation showing how the inhomogeneous ambient
magnetic field, the nonuniform density, and quantum effects influence the Jeans instability in a
self-gravitating quantum dusty magnetoplasma. We also use the intersection method developed
by Omar Khayyam [99] to solve the cubic equation and obtain minimum threshold condition
for Jeans instability.
4.2 Dielectric Response Function
We consider an infinitely extended inhomogeneous high density dusty magnetoplasma con-
taining electrons, ions and charged dust grains in the presence of an inhomogeneous ambient
magnetic field B0. We choose the Cartesian coordinate system in such a way that B0(x) is in
the z-direction with the inhomogeneity along x-direction and the wave propagation vector k is
in yz-plane. In order to satisfy the equilibrium conditions, given in Appendix A, we assume
that the streaming velocity v0α is along y-axis and the density inhomogeneity along x-axis.
Further, the charge quasi-neutrality condition is n0e(x) = n0i(x) +Zd n0d(x) where n0α(x) is
the equilibrium inhomogeneous number density of α-species (α = e, i, d), Zd is the dust charge
55
state. The density inhomogeneity produces diamagnetic drifts, and the magnetic field inhomo-
geneity causes uniform streaming of ions and electrons (not for heavy dust particles i.e., v0d
= 0) with v0e,i = −c ∂B0(x)/∂x
4πqdnody. Such plasmas may exist in the interiors and environments
of astrophysical compact objects e.g., white dwarfs and neutron stars/magnetars, supernovae,
etc [111]. The governing linearized equations for electrostatic wave propagation in the quantum
hydrodynamic (QHD) model [101]-[105] for the electrons, ions and charged dust grains in the
presence of the inhomogeneous ambient magnetic field B0(x) are
mαn0α
∂v1α∂t
+ (v0α.∇)v1α
= n0αqα
E1 +
1
cvα ×B0
−KTFα∇n1α
−mαn0α∇ψ1 +
4mα∇∇2n1α
, (4.1)
and∂n1α∂t
+ v0α.∇n1α + v1α.∇n0α + n0α∇ · v1α = 0. (4.2)
Poisson’s equations for the perturbed electrostatic potential φ1 and gravitational potential
ψ1 are
∇2φ1 = −4π
qαn1α, (4.3)
and
∇2ψ1 = 4πGmαn1α, (4.4)
where subscript 0 indicates equilibrium quantities while 1 is used for the perturbed quanti-
ties, summation over α is for the three species i.e., (α = e, i, d), is Planck’s constant divided
by 2π and qα,mα, c and G are the charge, mass, the velocity of light in vacuum and gravi-
tational constant respectively. Here, we may take into account the quantum effects of all the
species when they are considered extremely cold. In Eq. (4.1), we assume that the plasma
particles in a zero-temperature Fermi gas satisfying the pressure pFα = mαv2Fαn1α/3n20α where
vFα = (2kBTFj/mj)1
2 is the Fermi speed; kB and TFj are the Boltzmann constant and Fermi
temperature respectively.
56
By assuming that perturbations have sinusoidal character i.e.,
(E1, B1,n1α,v1α) ≈ e− i ω t+ i k·r , (4.5)
we may rewrite the equations (4.1-4.4) as
v1α =qαk φ1ω∗mα
+i
ω∗v1α ×Ω0α +
kv′2Fαω∗
n1αn0α
+k
ω∗ψ1, (4.6)
n1αn0α
=
k · v1αω∗
+
ivx1αLαω∗
, (4.7)
k2φ1 = 4π
q0αn1α, (4.8)
k2ψ1 = −4πGmαn1α. (4.9)
Here, ω and k are the angular frequency and wavenumber vector, respectively. We have also
defined
v′2Fα = v2Fα +2k2
4m2α
, Ω0α =qαB0mαc
z and ω∗ = ω − kyv0α .
Using Eqs. (4.6-4.9) and after some straightforward calculations, we obtain the general dielectric
susceptibility for a dusty magnetoplasma with magnetic field and density inhomogeneities as
χα = −ω2pα
'k2z
(ω∗)2+
k2y
(ω∗)2 −Ω20α
1− Ω0α
ω∗kyLα
(
k2 −'
k2z(ω∗)2
+k2y
(ω∗)2 −Ω20α
1− Ω0α
ω∗kyLα
(k2v′2Fα − ω2Jα
, (4.10)
where
ω2pα =4πn0αq
2α
mα, ω2Jα = 4πGmαn0α and Lα = − n0α
∂n0α/∂x.
Using the above equation (4.10), the electrostatic dielectric response function is given by
57
ǫ (ω, k) = 1 +
α
χα = 1−
α
ω2pα
'k2z
(ω∗)2+
k2y
(ω∗)2 −Ω20α
1− Ω0α
ω∗kyLα
(
k2 −'
k2z(ω∗)2
+k2y
(ω∗)2 −Ω20α
1− Ω0α
ω∗kyLα
(k2v′2Fα − ω2Jα
.
(4.11)
Eq. (4.11) is the electrostatic dielectric response function which incorporates the effects of
inhomogeneities, quantum effects through both the Fermi potential and the Bohm potential and
the effect of gravitational potential. On ignoring the inhomogeneities, this function immediately
reduces to the previously derived response function [28].
In the following, we shall derive the dispersion relation for the self-gravitational electrostatic
drift waves including the quantum effects for the electron dynamics, density inhomogeneity for
electrons and ions and the gravitational effects for dust grains. The quantum effects on ions
and dust grains are neglected due to their heavier masses. Also, being insignificantly small, we
can neglect the gravitational effects on electrons and ions. However, for dust grain, the self-
gravitational effect is taken into account. Thus for such plasmas satisfying the above conditions
and assuming that the wave frequency and Doppler shifted frequency is much less than the gyro
frequency of electron i.e., ω, ω∗ << |Ω0e| and the phase velocity of the wave is less than the
effective electron (quantum) velocity but greater than the ion one i.e.,v′Fi << ω/kz << v′Fe ,
we can write the dispersion relation of electrostatic waves i.e., ǫ(ω, k) = 0 by simplifying Eq.
(4.11) as
1 +ω2pe
k2v′2Fe−
ω2pik2
k2z
(ω∗)2+
k2y
(ω∗)2 −Ω20i
1− Ω0i
kyω∗Li
−
ω2pd
k2zω2
+k2y
ω2 −Ω20d
k2 +
k2zω2
+k2y
ω2 −Ω20d
ω2jd
= 0.
(4.12)
The condition for the phase velocity (i.e., v′Fi << ω/kz << v′Fe) defines that electrons which
run along magnetic lines will reach thermal equilibrium condition quickly whereas ions can not
reach thermal equilibrium and should be described by the drift equation.
In the following, we will discuss two frequency domains i.e., Ω0d << ω < Ω0i for unmagne-
tized dust and ω << Ω0d < Ω0i for magnetized dust grains.
58
4.2.1 For Unmagnetized Dust Grains
For this case, we shall assume the intermediate frequency domain i.e., the wave frequency lies
between the gyro frequencies of dust and ions (Ω0d << ω < Ω0i) . Thus the dispersion relation
reduces to
1 +ω2pe
k2v′2Fe−
ω2pik2
k2z
(ω∗)2−
k2yΩ20i
1− Ω0i
kyω∗Li
−
ω2pdω2 + ω2jd
= 0. (4.13)
If we assume that the propagation is predominantly perpendicular i.e., k2z << k2y, the above
equation takes a simpler form
1 +ω2pe
k2yv′2Fe
+ fi −ω′iω∗−
ω2pdω2 + ω2Jd
= 0, (4.14)
where
ω′i =ω2pi
Ω0iLikyand fi =
ω2piΩ20i
.
From Eq. (4.14), we notice that the drift wave may become unstable due to the presence
of dust particles satisfying the certain threshold conditions for both the inhomogeneous plasma
(i.e., ω′i = 0) and the homogeneous (i.e., ω′i = 0). In the absence of dust particles, we only
obtain the real oscillations in the former case while in the latter no wave would exist. We
also note that for the plasma system satisfying above assumptions (see Appendix F. and Eq.
(4.14)), the effect of streaming will appear only if the inhomogeneities in both the ambient
magnetic and the equilibrium density are present.
We may rewrite the above Eq. (4.14) as
0 = ω3 −kyv0 + ω′i/fiF
ω2 + ω
ω2Jd − ω2dlh/F
−)
kyv0 + ω′i/fiF
ω2Jd − ω2dlh/F+ω2dlh/F
ω′i/fiF
*, (4.15)
where
F = 1 +1
fi+
ω2pefik2yv
′2Fe
, ωdlh =ωpdΩ0iωpi
.
The above equation becomes cubic in ω due to inhomogeneities and gives three roots. We
observe that one root remains always real where as the other two roots can become complex
59
if the discriminant of the cubic equation becomes negative. One of them would give the
instability called Jeans instability for inhomogeneous streaming dusty magnetoplasma satisfying
the threshold condition. The second root will give damping with negative real frequency which
is not physical and thus we ignore it. In order to get the exact minimum threshold condition
for Jeans instability, we use intersection method [99]. We split this cubic equation into a cubic
and a parabolic function and then find out the intersection points for roots. The mathematical
formulation of this intersection method is given in the Appendix. E. We note that the instability
would occur only if the y-component of the vertex point of parabola (i.e. minima ) is greater
than zero. Thus the resultant threshold condition in Eq. (F.10) is given by
ω2Jd − ω2dlh/F
> 2
kyv0 + ω′i/fiF
2
1 +
ω2dlh/F
(ω′i/fiF )
(kyv0 + ω′i/fiF )3− 1
. (4.16)
The above threshold condition contains modification due to the the inhomogeneities in the
equilibrium density and in the ambient magnetic field. It is evident that both the density
inhomogeneity and the streaming velocity not only reduce the growth rate of Jeans instability
but also give rise to real oscillations. Similarly the quantum effects through both the Fermi
and the Bohm potentials also tend to stabilize the instability. Further, we observe that the
gravitational potential reduces the real oscillations and that the ion streaming velocity v0i and
the effective quantum velocity v′Fe causes the drift waves to propagate perpendicular to the
ambient magnetic field. The graphical representation to aid the instability threshold is given in
Fig (5.1) for typical parameters [111] (cgs system of units) for the interiors of the neutron stars,
magnet stars, and white dwarfs, me = 9.0×10−28 g , mi = 12 mp (mp = 1.672×10−24g),md ∼1015 mi, noe ≃ noi ∼ 1027cm−3, nod = 10−6noi, Z = 103, Le = Li ∼ 1000 cm and Bo = 109G.
If we assume that the last term in curly brackets of cubic Eq. (4.15) becomes negligibly
small and thus the equation reduces to
0 = ω3 −kyv0 + ω′i/fiF
ω2 + ω
ω2Jd − ω2dlh/F
−)
kyv0 + ω′i/fiF
ω2Jd − ω2dlh/F*
,
(4.17)
60
Figure 4-1: This figure represents the graph of f(ω) vs ω for Eq. (4.15). Solid curve is forf(ω) = ω3 and dotted curves are for f(ω) = B ω2−ω C + G for set of parameters given afterEq. (4.16) with variation of streaming velocity as (i) dotted for v0 = 10−7c, (ii) small dashedfor v0 = 3× 10−7c (iii) large dashed for v0 = 5× 10−7c.
61
which may be factorized as
)ω2 +
ω2Jd − ω2dlh/F
*)ω −
kyv0 + ω′i/fiF
*= 0. (4.18)
The first root gives the growth rate of Jeans instability as
γ =ω2Jd − ω2dlh/F
12 , (4.19)
with the threshold condition
ω2Jd − ω2dlh/F > 0,
or ωpeωpd
ωdlh
ω2dlhω2Jd
−
1 +Ω20iω2pi
12
≥ kyv′Fe. (4.20)
The second root gives real oscillations propagating with the streaming and the Fermi velocities
as
ωr = kyv0 + ω′i/fiF. (4.21)
Eqs. (4.19, 4.21) show that due to the presence of inhomogeneities, the real propagating
oscillatory behavior is also observed with the Jeans instability. We observe that the dust-lower-
hybrid wave and the quantum mechanical effects of electrons tend to reduce the growth rate.
We also note that real frequency of self-gravitational drift waves increases with the increase in
both the inhomogeneities and the quantum effects through the Fermi and the Bohm potentials.
The expression for the growth rate in Eq. (4.19) is different from the previously derived
result [28] as it contains the contribution of quantum effects of electrons instead of dust grains.
Thus as a result, we obtain a new threshold condition on wavenumber for the instability to
occur. For the chosen parameters, the general threshold condition given in Eq. (4.16) also
immediately reduces to the above condition given in Eq. (4.20) because the second term in the
square root becomes negligibly small.
Eq. (4.15) is plotted in Fig.(4.1) for the above set of parameters. We observe that
62
the real frequency increases with the increase of streaming velocity. We also note that for
these parameter the values of the real frequency fulfill the condition of Ω0d << ω < Ω0i.
For homogeneous plasma for which ω′i → 0, Eq. (4.14) becomes
ω2 = −ω2Jd + ω2dlh/F , (4.22)
giving purely growing Jeans instability as
γ =ω2Jd − ω2dlh/F
12 . (4.23)
If the gravitational potential is absent, Eq. (4.14)
1 +ω2pe
k2yv′2Fe
+ fi −ω′iω∗−
ω2pdω2
= 0,
or
1− (ω′i / fiF )
ω∗− ω2dlh /F
ω2= 0. (4.24)
This is the same dispersion relation as derived by Salimullah et al. [100] and by letting
ω = kyv0i + δ where δ << kyv0i, the above equation becomes
1− (ω′i / fiF )
δ− ω2dlh /F
k2yv20i
1− 2δ
kyv0i
= 0.
The growth rate of drift wave is given by (ω = ωr + iγ)
γ =
|ω∗i | k3yv30√
2ωpd
+,,-1−ω2pd
8 |ω∗i | kyv0
k2yv
20
ω2pd/F′− 1
2, (4.25)
where
F ′ = 1 +ω2pek2v′2te
+ fi/
This growth rate of drift wave is more general with the threshold condition
63
|ω∗i | >ωpd
2√
2kyv0
k2yv
20
ω2pd/F′− 1
, (4.26)
If we assume kyv0i ∼ ωpd /√F ′, it immediately reduces to the result of Salimullah et al.
[100] i.e.,
γ =ωpi (kyv0i)
3
2
√2ωpd
kyΩ0i |Li|
. (4.27)
4.2.2 For Magnetized Dust Grains
If the dust is also magnetized, the dispersion relation Eq.(4.12) takes the form
1 +ω2pe
k2v′2Fe−
ω2pik2
k2z
(ω∗)2−
k2yΩ20i
1− Ω0i
kyω∗Li
−
ω2pd
k2zω2−
k2yΩ20d
k2 +
k2zω2−
k2yΩ20d
ω2jd
= 0. (4.28)
For perpendicular propagation, (i.e., kz = 0) , the above equation reduces to
1− (ω′i / fiF )
ω∗+
ω2dlh /F
Ω20d − ω2jd
= 0, (4.29)
or
ω = kyvy0i +
(ω′i / fiF )1 +
ω2dlh /F
Ω20d − ω2jd
. (4.30)
From the above equation it is evident that for the magnetized dust grains, we only obtain
stable wave and that the frequency reduce with the increase in the gravitational potential.
64
Chapter 5
Summary of Results and Discussion
In Chapter 2, the drift kinetic theory is used to derive the full kinetic dispersion relation
for the Geodesic acoustic modes (GAMs) including diamagnetic effects due to inhomogeneous
plasma density and temperature. The fluid model, including the effects of ion parallel viscosity
(pressure anisotropy) is used to recover exactly the adiabatic index obtained in kinetic theory.
GAM results from the balance between the ion polarization current (Eq. (2.45)) and the
geodesic current (Eq. (2.46)), which is produced by the M = ±1 side-bands of the perturbed
pressure (together with parallel viscosity [27]). The pressure side-bands are in turn created by
the flow due to M = 0 mode of the electrostatic potentia l[16]. As pointed out by [35], GAMs
are degenerated in the absence of diamagnetic effects in the q >> 1 approximation. Drift effects
directly affect the diamagnetic current, (Eq. (2.43)), breaking the symmetry of the dispersion
relations (Eqs.(2.47,2.48)) removing the degeneracy of the zero frequency zonal flow mode.
The instability occurs for high temperature gradients ηi > 3/4. The grow rate of this
instability, which is associated with the zonal flow branch, as well as the modification of the
GAM frequency are typically small for density/temperature gradients length scale of the order of
the minor radius, i.e., γ ∼ (ρiR0/rLn)ωgam. However, in the region of large density/temperature
gradients, the drift corrections may become of the similar order as the GAM frequency.
To our knowledge this instability of the ZF/GAM type mode has not been studied before,
though we have shown that our final dispersion relation is a special case of a more general
fishbone-like dispersion relation obtained by Zonca et al., [50]. The fluid model presented in
65
this section provides a transparent treatment of the diamagnetic drift corrections and thus a
shorter path to the understanding of the physical mechanism. The fluid treatment is also more
easily generalized to the nonlinear case. One has to note that technically, the fluid treatment
requires the condition ω > vti/qR0, therefore our results are valid either in high q regimes, or
in case of large plasma gradients, so that ω∗ > vti/qR0. In stable case, ηi < 3/4, the dispersion
relation (2.48), produces two GAM type modes, which, in the region of high density gradient,
can have comparable frequencies. It is of interest to note that in experiments, there are two
modes with close frequencies observed [51],[52]. One can conjecture that this mode splitting is
due to plasma gradients effects.
Previously, the effect of equilibrium (poloidal and toroidal) rotation on GAM were inves-
tigated within the framework of one fluid MHD theory in [53],[54]. It was found that the
equilibrium plasma rotation induces not only the sin θ, but also cos θ component of the per-
turbed density (and pressure); θ is the poloidal angle in toroidal configuration with equilibrium
magnetic field B = B0(1 − ǫ cos θ), ǫ = r/R0, where r is the minor and R0 are major radii.
In this respect, density gradients have similar effect as equilibrium rotation [53], [54], caus-
ing the perturbed density to depend also on cos θ (Eq. (2.34)) and φc = 0 (Eq. (2.37)), i.e.,
ni = O(k2rρ2i ) + O(krρi) sin θ +O(krρi) cos θ and φ = φ0 + φs sin θ + φc cos θ. Higher order
of dispersive terms, O(k4rρ4i ), which are important to understand the radial structure of GAM,
although not included in this paper, can be obtained by accounting for φ±2.
Similar dispersion relation for electromagnetic modes including dispersive terms were derived
in [38], [39] (see also [18], [35]). It is expected that, generally, φk ∼ (krρi)kφ0 (Eq. (2.37)), ni0 ∼
k2rρ2i
eφ0/Ti
, ni±1 ∼ krρi
eφ0/Ti
(Eq. (2.34)) [31], and, hence, as confirmed previously[16]
the main quantities involving in the basic GAM study are φ0 and ρs sin θ (mass density).
In Chapter 3, we have used the drift kinetic equation to derive the general dispersion
relation for GAMs which incorporates the effect of magnetic perturbation. We have derived
general dispersion relation for arbitrary values of se = ωqR/vTe and si = ωqR/vTi. The
latter includes the dispersive as well as damping effects. The simplified plasma response in
the electron adiabatic (se < 1) and fluid ion (si > 1) regimes were also obtained showing the
magnetic effect explicitly. The main result is that the GAM have finite magnetic component
in the first side-band contrary to the result in Refs. [69, 73]. It is worth noting that the
66
amplitude of the magnetic vector potential does not contain the additional small parameter
krρi as suggested in Refs. [69, 73], but is controlled by the value of the βeq2 parameter. For
βeq2 ≃ 1, the amplitude of the magnetic vector potential is of the same order as the main
GAM sideband φ1s [see Eq. ((3.49))]. We note that βeq2 parameter can be large as e.g., in
NSTX tokamak[55]. The dependence of GAM frequency on βe is shown in Figs. (3.4)-(3.6).
It was recently noted that the value of the βeq2 parameter also responsible for the coupling of
m = n = 0 mode to the m = 2 harmonics [75]. The effect investigated in our paper is different
since it is concerned with the magnetic component of the m = 1 harmonics. Essentially, it is
a two-fluid effect since we assume different regimes for the electron and ion responses: se < 1
and si > 1. It is interesting that in the lowest order, the GAM dispersion relation is largely
insensitive whether the m = 1 side-band is electrostatic or electromagnetic. It may be noted
that the transition to the electromagnetic regime requires the large radial mode width, K2⊥ < 1;
K2⊥ ≃ τk2rc
2/q2ω2pi
≃ τk2rρ
2i /βiq
2. The theoretical dependencies of the amplitude of the
magnetic and side band harmonics shown in Eqs. (3.49), (3.50), can be used for experimental
identification of these effects.
The effects of plasma pressure βe, the safety factor q and the temperature ratio τ on GAM
dispersion have been analyzed here. We note that the dispersion is negative for GAMs in
some regions and this dispersion shifts to lower (higher) frequency with the increase in the
plasma pressure βe and safety factor q (temperature ratio τ). The full study of the dispersion
effects requires the inclusion of the second harmonics. We outline full route to the calculations
of the second harmonics effects and the differences with previous work in Appendix E. The
investigation of the dispersive effects (up to the second order) as well as the mode damping
[45] in full electromagnetic model is left for future work. It is expected that the amplitude
and the phase relations for plasma parameters fluctuations may be important for experimental
identification of GAM magnetic m = 1 component.
In Chapter 4, The stability analysis of self-gravitational electrostatic drift waves for a
streaming non-uniform quantum dusty magnetoplasma by using the quantum hydrodynamic
model of plasmas is presented. Incorporating the effects of density inhomogeneity∇n0α, stream-
ing velocity v0α due to magnetic field inhomogeneity ∇B0, Bohm potential and the Fermi
degenerate pressure, we first derive the general dispersion relation for self-gravitational electro-
67
static drift waves and then simplify it for two different frequency domains i.e., Ωd << ω < Ωi
(unmagnetized dust) and ω << Ωd < Ωi (magnetized dust).
For both frequency domains, the effects of density inhomogeneity give rise to real oscilla-
tions and the ion streaming velocity v0i and the effective quantum velocity provide the source
to propagate these oscillations perpendicular to the ambient magnetic field. This oscillatory
behavior of self-gravitational electrostatic waves increases with increase in inhomogeneities and
decreases with increase in quantum effects through both the Bohm potential and fermi potential
and in gravitational potential.
We also note that for the magnetized case, the available free energy through density inhomo-
geneity and the streaming velocity is not sufficient to make the drift waves unstable. However
only for the unmagnetized case, the electrostatic drift waves may become unstable under appro-
priate conditions. These unstable self-gravitational drift waves give rise to Jeans instability and
by using the intersection method for solving cubic equation, we also determine the threshold
condition for Jeans instability. We also note that the inhomogeneity in the ambient magnetic
field ( equilibrium density) through streaming velocity (diamagnetic drift velocity) suppress the
Jeans instability depending upon the characteristic scale length of these inhomogeneities. On
the other hand, the dust-lower-hybrid wave and the quantum mechanical effects of electrons
tend to reduce the growth rate.
For the unmagnetized case, we further simplify the dispersion relation for growth rate in
some limiting cases. e.g., in the absence of gravitational potential, we obtain a new dispersion
relation for the growth rate of drift wave instability in non-uniform quantum dusty magneto-
plasma which incorporates the quantum effects through electrons. We also retrieve the growth
rates of Jeans instability in a homogeneous dusty magnetoplasma.
Our results in this paper may be useful for the study of Jeans instability and the possible
drift waves for the nonuniform streaming dusty quantum plasmas which may occur e.g., in
dense astrophysical systems i.e., the interiors of white dwarfs and neutron stars [91, 92, 93],
laser-produced plasmas [94, 95, 96, 97] and in the laboratory plasmas[89, 90].
68
Chapter 6
Appendixes
6.1 Appendix A
The required integrals for the derivation of general dispersion relation are
Γ0α =1
n0
J20 (zα)FMαd
3v (A.1)
Γ0α =
1
n0
J20 (zα)
v2
v2tαFMαd
3v (A.2)
I00α =1
n0
ω2
WJ20 (k⊥v⊥/ωci) FMαd
3v (A.3)
I01α =1
n0
ω2
WJ20 (k⊥v⊥/ωci)
v2
v2tαFMαd
3v (A.4)
I02α =1
n0
ω2
WJ20 (k⊥v⊥/ωci)
v4
v4tαFMαd
3v (A.5)
I10α =1
n0
v2⊥/2 + v2
v2tα
ω2
WJ20 (k⊥v⊥/ωci) FMαd
3v (A.6)
I11α =1
n0
v2⊥/2 + v2
v2tα
v2
v2tα
ω2
WJ20 (k⊥v⊥/ωci) FMα d3v (A.7)
I20α =1
n0
v2⊥/2 + v2
2
v4tα
ω2
WJ20 (k⊥v⊥/ωci) F0 d3v (A.8)
I21α =1
n0
v2⊥/2 + v2
v2tα
2v2
v2tα
ω2
WJ20 (k⊥v⊥/ωci) FMαd
3v (A.9)
69
and the drift integrals are
I00∗ =1
n0
Qα
ω2
WJ20 (k⊥v⊥/ωci) FMαd
3v = I00α + ηαP00α (A.10)
P 00α =1
n0
ω2
W
v2
v2Ti− 3
2
J20 (k⊥v⊥/ωci) FMαd
3v (A.11)
I10∗α =1
n0
QαQ
v2⊥/2 + v2
v2Ti
ω2
WJ20 (k⊥v⊥/ωci) FMα d3v = I10α + ηαP
10α (A12)
P 10α =1
n0
v2⊥/2 + v2
v2Tα
v2
v2Tα− 3
2
ω2
WJ20 (k⊥v⊥/ωci) FMαd
3v (A13)
I20∗α =1
n0
Qα
v2⊥/2 + v2
2
v4Ti
ω2
WJ20 (k⊥v⊥/ωci) FMα d3v = I20α + ηαP
20α (A.14)
P 20α =1
n0
v2⊥/2 + v2
2
v4Ti
v2
v2Ti− 3
2
ω2
WJ20 (k⊥v⊥/ωci) FMα d3v (A.15)
with
Qα = 1 + ηα
v2
v2Ti− 3
2
(A.16)
70
6.2 Appendix B
Assuming the low drift frequency and the small Larmor radius (i.e., ω2 >> ω2de and J20 (k⊥v⊥/ωce) ≃1) for electron case, we may rewrite W ≃ ω2 − v2/q
2R2 and integrals become
Γ0e = 1 (B.1)
Γ0e =
1
2(B.2)
I00e = −seZ(se) (B.3)
I01e = −s2eZ1(se) (B.4)
I02e = −s2e1
2+ s2eZ1(se)
= −s2e
1
2− I011
(B.5)
I10e = −1
2seZ(se) + s2eZ1(se)
=
I00e2− I01e (B.6)
I11e = −s2e1− I10e
(B.7)
I20e = −s2e2
+1
2I00e + (1 + s2e)I
01e (B.8)
I21e = I11e − s2e
3
4+ s2eZ3(se)
(B.9)
where
∞
−∞
dx exp[−x2]
π1/2x2 − s2e
=1
seZ(se) ; se =
ωqR
vte(B.10)
∞
−∞
x2dx exp[−x2]
π1/2x2 − s2e
= 1 + seZ(se) = Z1(se) (B.11)
∞
−∞
x4dx exp[−x2]
π1/2x2 − s2e
=1
2
1 + 2s2eZ1(se)
= Z3(se) (B.12)
∞
−∞
x6dx exp[−x2]
π1/2x2 − s2e
=3
4+ s2eZ3(se) (B.13)
expanding the plasma dispersion function for small argument i.e., s2e << 1,
Z(se) = i√πexp
−s2e
− 2se
1− 2
3s2e +
4
15s4e −
8
105s6e
(B.14)
71
we may reduce the integrals electron expressions. In what follows we neglect the imaginary
part
I00e = 2s2e −4
3s4e (B15)
I01e = −s2e + 2s4e (B16)
I02e = −s2e2− s4e (B17)
I10e =4
3s4e −
16
15s6e (B.18)
I11e = −s2e +4
3s6e (B.19)
I20e = −s2e2
+s4e3
(B.20)
I21e = −7
4s2e −
s4e2
(B.21)
As we consider the ηe is negligibly small, so that the result of integrals (A.10-A.15) becomes
similar to the above integrals.
6.3 Appendix C
The ion expressions are calculated with the expansion
ω2
W= 1 +
v2ω2q2R2
+ω2di2ω2
+v2ω
2di
q2R2ω4(C.1)
and
J20 (zi) = 1− 1
2
k2rv2⊥
Ω2i(C2)
72
Γ0i =1
n0
J20 (k⊥v⊥/ωci)FMid
3v = 1− 1
2k2rρ
2i (C.3)
Γ0i =
1
n0
J20 (k⊥v⊥/ωci)
v2
v2tiFMid
3v =1
2− k2rρ
2i
4(C.4)
I00i =1
n0
ω2
WJ20 (k⊥v⊥/ωci) FMid
3v
= 1− 1
2k2rρ
2i +
1
2s2i+
7
8
ω2diω2
+23
8
ω2dis2iω
2(C.5)
I01i =1
n0
ω2
WJ20 (k⊥v⊥/ωci)
v2
v2tiFMid
3v
=1
2− k2rρ
2i
4+
3
4s2i+
23
16
ω2diω2
+141
16
ω2dis2iω
2(C.6)
I02i =1
n0
ω2
WJ20 (k⊥v⊥/ωci)
v4
v4tiFMid
3v
=3
4− 3
8k2rρ
2i +
15
8s2i+
141
32
ω2diω2
+1185
32
ω2dis2iω
2(C.7)
I10i =1
n0
v2⊥/2 + v2
v2ti
ω2
WJ20 (k⊥v⊥/ωci) FMid
3v
= 1− 3
4k2rρ
2i +
1
s2i+
9
4
ω2diω2
+87
8
ω2dis2iω
2(C.8)
I11i =1
n0
v2⊥/2 + v2
v2ti
v2
v2ti
ω2
WJ20 (k⊥v⊥/ωci) FMi d
3v
= 1− 5
8k2rρ
2i +
9
4s2i+
87
16
ω2diω2
+171
4
ω2dis2iω
2(C.9)
I20i =1
n0
v2⊥/2 + v2
2
v4Ti
ω2
WJ20 (k⊥v⊥/ωci) FMi d
3v
=7
4− 13
8k2rρ
2i +
23
8s2i+
249
32
ω2diω2
+1641
32
ω2dis2iω
2(C.10)
I21i =1
n0
v2⊥/2 + v2
v2ti
2v2
v2ti
ω2
WJ20 (k⊥v⊥/ωci) FMid
3v
=23
8− 33
16k2rρ
2i +
141
16s2i+
1641
64
ω2diω2
+15867
64
ω2dis2iω
2(C.11)
with ρi = vti/Ωi and si = ωqR/vti.
The drift integrals for ions by using the assumptions used in the case of drift effects on
GAMs become as
73
I00∗i =1
n0
Qi
ω2
WJ20 (k⊥v⊥/ωci) FMαd
3v ≃ 1 +1
2s2i(1 + 2ηi) (C.12)
I10∗i =1
n0
Qi
v2⊥/2 + v2
v2Ti
ω2
WJ20 (k⊥v⊥/ωci) FMα d3v ≃ 1 + ηi +
1
s2i(1 + ηi) (C.13)
I20∗α =1
n0
Qα
v2⊥/2 + v2
2
v4Ti
ω2
WJ20 (k⊥v⊥/ωci) FMα d3v =
7
4+
23
8s2i+
ηis2i
(C.14)
6.4 Appendix D
Using the expansions of electrons and ions, we may simplify the coefficients as
S01 = Γ0i − 1 +1
2K2 +
1
2K2 (D.1)
S02 = 1 + 2K (D.2)
S03 = S02 (D.3)
S1 = −S01 (D.4)
S2 = K − τ−1 (D.5)
D1 = S2 (D.6)
Resultantly, we obtain
S1S2D1
− S2 +D1 = S1 (D.7)
S03S02
S2D1
− 1 = 0 (D.8)
ωdiω
S02 = K1 +K1 (D.9)
Where
K1 =ωdiω
, K2 =7
4
ω2diω2
, K =1
2s2i, K1 =
ωdiω
1
s2iand K2 =
23
8
ω2dis2iω
2
74
6.5 Appendix E
Here we describe the differences of our approach with that of the Refs. [69, 73]. Ref. [69]
employs the perturbative solution of the drift kinetic equation which corresponds to the zero
Larmor radius limit of the gyrokinetic equation (2) used in our work, zα → 0, J20 (zα) → 1. As
result, the finite Larmor radius effects, which are of the same order as magnetic drift effects
(or the Finite Orbit Width effects, in terminology of Ref. [69], k2rρ2i ≃ ω2d/ω
2 are neglected
in Ref. [69]. The second important point is neglect of the coupling to the zeroth harmonic of
the distribution function in Eqs. (1a, 1b) of Ref. [69], is described by the first terms in Eqs.
(3.5,3.7), see also Eqs. (E.8-E.10). As a result, a number of the second order terms are missing
in Ref. [69]. Namely, our expression (3.13) for f1s has the structure similar to Eq. (2b) of
Ref. [69] but with the following corrections: the factor in front of the φ1s term in Ref. [69] is
replaced in our work with
v2
v2 − (qRω)2→
v2 + ω2d/2q2R2ω2
v2 − (qRω)2 + ω2d/ (2q2R2ω2)J20 (zα). (E.1)
Similarly, the factors in all other terms in Eq. (2b) [69] should be replaced as
1
v2 − (qRω)2→ 1
v2 − (qRω)2 + ω2d/ (2q2R2ω2)J20 (zα). (E.2)
The resonant factors in Eq. (2c) for f1c [69] should also be replaced:
1
v2 − (qRω)2→ 1
v2 − (qRω)2 + ω2d/ (2q2R2ω2)J20 (zα). (E.3)
Note that there is a number of typos in Eq. (2c); after these typos are corrected, this equation
becomes similar to our Eq. (3.12) with the above noted factors replacements (Eqs. (E.1-
E.3). Therefore the terms of the k2rρ2i and ω2d/ω
2 are incompletely included in Ref. [69].
The n1s density component expression in Ref. [69] is missing theΓ0i − 1 + 1
2K2 + 1
2K2
φ1s
term (compare with our Eq. (3.35)). It is this last term which is responsible for a finite
value of the first harmonic of parallel current and magnetic field perturbation. It can be of
the same order as the retained term with K, since most generally, K ≃ k2rρ2i < 1. The
75
neglect of the coupling to f0 becomes critical for correct calculation of the second harmonics of
the ion distribution function (and density), which are of the second order with respect to φ0,
n2s,c/n0 ≃ ω2d/ω2 (eφ0/Ti). As shown in Appendix E, in this case the factor
v2 − (qRω)2
−1
has to be replaced with the function D given by Eq. (E.10) which produces additional terms
of the order of ω2d/ω2 missed in Ref. [69] .
Alternatively, the kinetic effects in GAM could be investigated via the formal solution of the
gyrokinetic equation in the form of infinite series of Bessel functions [74, 68, 45]. This approach
was used in Ref. [73] to investigate the electromagnetic and second order effects. The starting
point is the same equation (3.2) which is written as
ω + ωdα sin θ +
ivqR
∂
∂θ
gα = ωJ20 (zα) (φ−
vcA)
eαFMα
Tα. (E.4)
This equation can be formally integrated giving the solution in the form [74, 68, 45]
ga =eαFMα
Tα
∞
l=−∞
J20 (krv⊥/ωcα)∞
m,n=−∞
im−n exp (i (m− n) θ)
×ωJm
ωdαqR/v
JnωdαqR/v
ω − (n− l) v/qR(exp (ilθ)φl −
vcAl exp (ilθ)) (E.5)
This expression is a basis of the method in Ref. [73]. In this form the contribution of the
harmonics of poloidal sidebands φ±1, φ±2, .., can be represented by the product of two infinite
series, one in ωdαqR/v and the other in v/ωqR expansions. This basis, though formally correct
is not consistent with natural expansion parameters for GAM modes: k2rρ2i < 1 and ω2d/ω
2 < 1.
The basic GAM ordering follows from the balance of the radial polarization current against
the radial diamagnetic current
k2rρ2s
eφ0Te
≃ ωdω
p±1p0
. (E.6)
The pressure oscillations are determined by the finite compressibility of poloidal plasma flow
due to the inhomogeneous magnetic field
ωp±1p0
≃ ωdeφ0Te
. (E.7)
This balance determines the GAM frequency ω2 ≃ ω2d
k2rρ2s
≃ τev2Ti
R2. It is easy to see that natural
76
expansion for higher order (dispersive) corrections to GAM frequency is based on the expansion
in two parameters: k2rρ2i and ω2d/ω
2, and their products. It is important to note that the basic
GAM balance in Eqs. (E.6, E.7) naturally define that these two expansion parameters are of the
same order. The contribution of the ion sound effect is determined by the subsidiary expansion
in v/ (ωqR) parameter, while the Landau damping corresponds to the maximal ordering ω ≃v/qR. In our approach, presented in Eqs. (3.5-3.7), following Refs. [17, 78], the expansions
in ω2d/ω2 and v/ (ωqR) can be traced and explicitly written down via the expansion of the W
function in Eqs. (3.8-3.10). The expansion in k2rρ2i is multiplicative and follows from the J20 (zα)
function in Eqs. (3.8-3.10).
In the approach of Eq. (E.5), there are two infinite series involving the products of the Bessel
functions: JmωdαqR/v
JnωdαqR/v
and resonance denominator
ω − (n− l) v/qR
−1.
The latter is responsible for Landau damping resonances. It is known however that in toroidal
geometry the Landau resonance is modified by the ωd corrections [76, 77]. This is clearly seen
from the structure of the W function in our Eqs. (3.8-3.10), W = ω2 − v2/q2R2 − ω2dα/2. In
the fluid limit ω ≫ v/qR, the resonancesω − (n− l) v/qR
−1are removed by the expansion
using v/ (ωqR) as a small parameter. The natural small parameter ω2d/ω2 ≪ 1, existing for
GAM, does not help to deal with the infinite sum of JmωdαqR/v
JnωdαqR/v
. To make
it tractable, one has to assume an auxiliary small parameter ωdαqR/v < 1 and expand the
Bessel functions retaining few lowest order terms. This however generates some terms which
have singularity at v = 0, JmωdαqR/v
≃ωdαqR/2v
m.
In general case, ωdαqR/v parameter is not small, ωdαqR/v ≃ 1, and one has to deal with
infinite series of JmωdαqR/v
JnωdαqR/v
. For example, it is not clear how to reproduce
in Eq. (E.5) the pure toroidal resonance ω2 − ω2dα/2 = 0 which formally follows in the limit
q →∞. It is interesting that the contribution of the first harmonics φ±1, A±1 in equation (E.5)
is equivalent to our expressions (10-12) and has no singularities. However, the second order
terms with φ±2, A±2 are singular at v = 0, e.g., see the explicit expressions for the second
order terms in Ref. [45]. The singular second order terms were not considered in Ref. [73]
which cast doubts on its accuracy.
The above difficulty is avoided in our approach, which can be straightforward generalized
77
to include the second harmonics. The corresponding equations can be written in the form
D ·X =Ψ, (E.8)
where the vector X =(g0, g1c, g1s, g2c, g2s), the vector
Ψ = ωeαFMα
TαJ20 (zα)
φ0
φ1c −vc A1c
φ1s −vc A1s
φ2c −vc A2c
φ2s −vc A2s
, (6.1)
and the matrix D
D =
ω 0 iωdα/2 0 0
0 ω −v/qR 0 iωdα/2
−iωdα −v/qR ω iωdα/2 0
0 0 −iωdα/2 ω −2v/qR
0 −iωdα/2 0 −2v/qR ω
. (E.10)
The structure of the toroidal and ion sound resonances becomes more complicated in this case
and is determined by the determinant of the matrix D
det |D|=ω5 − 5ω3v2/q2R2 − ω3ω2dα + ωω2dαv
2/q
2R2 + 4ωv4/q4R4 + 3ωω4dα/16. (E.11)
In neglect of ωd coupling one has
det |D| ≃ ωω2 − v2/q
2R2
ω2 − 4v2/q2R2
, (E.12)
which shows coupling of the first and second order resonances.
Full expressions for W and Eq. (E.11) demonstrate modification of the resonances by the
toroidal effects ωd, and offers a straightforward expansion in ωda/ω and v/ (ωqR) parameters.
It would be of interest to directly compare the results given by equations (E.8-E.10) up to the
78
second order. This is left for future work.
6.6 Appendix F
Zeroth Order equations
Assuming that there is no external electric field E0, constant streaming velocity v0 and
equilibrium density is space dependent only, we may write the momentum equation as
0 =qαc
(v0α ×B0)−∇PFαn0α
− ∇24n0αmα
∇n0α
(F.1)
The equation of continuity as
v0α ·∇n0α = 0 (F.2)
and the Maxwell curl equation as
c∇×B0 = 4π
n0αqoαv0α = 4πn0dZdv0 (F.3)
For choosing such frame in which electrons stream with relative velocity v0 with respect to ions
and dust being heavy is assumed as stationary (v0d = 0).
Taking cross product of Eq. (F.1) with B0, we obtain
v0 =c
en0e
B0 ×∇PFe
B20
(F.4)
The above set of equations describes the initial conditions of plasma system considered.
Eq.(F.4) shows that the streaming velocity (Diamagnetic drift) is perpendicular to both the
magnetic field and the density gradient and Eq. (F.3) shows that the magnetic field inhomo-
geneity is possible only if we consider the case of nonhomogeneous (density inhomgeneity) dusty
plasma. Thus our system contains both the
Instability Threshold
79
The Eq.(4.15) may be rewritten as
ω3 −B ω2 + ω C −G = 0 (F.5)
where
B =kyv0 + ω′i/fiF
, C =
ω2Jd − ω2dlh/F
, D =
ω2dlh/F
ω′i/fiF
and G = B C +D
The coefficients B and D are always positive for our considered plasma system. For Jeans
instability, the coefficient C is also positive. Therefore, all coefficients are kept positive in the
above equation.
Now we use the intersection method for a cubic function
f(ω) = ω3 (F.6)
and the parabolic function
f(ω) = B ω2 − ω C +G (F.7)
The above equation may be rewritten as a standard equation of parabola
ω − C
2B
2= 4
1
4B
y −
G− C2
4B
(F.8)
with the vertex point
(h, k) =
C
2B,G− C2
4B
We note that the instability will occur when the y-intercept of vertex point will be greater than
zero which is
G− C2
4B> 0 (F.9)
80
or
ω2Jd − ω2dlh/F
2−4kyv0 + ω′i/fiF
2 ω2Jd − ω2dlh/F
−4kyv0 + ω′i/fiF
ω2dlh/F
ω′i/fiF
< 0
By solving the above quadratic equation, one root gives the threshold condition
ω2Jd − ω2dlh/F
> 2
kyv0 + ω′i/fiF
2
1 +
ω2dlh/F
(ω′i/fiF )
(kyv0 + ω′i/fiF )3− 1
(F.10)
Here we have selected the one root which gives the Jeans instability and neglected the other
root because it gives the negative real frequency which is not physical.
81
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